Trigonometry

Contents

**Recall Triangle and its Types** 2

**Describe the Properties of Triangles** 3

**Convert the Angle Measurement of ° into Radians and vice-versa** 4

**Describe the Properties of Right Angle Triangles** 5

**Describe the Six Trigonometric Functions** 7

**Recall the Basic Trigonometric Ratios** 9

**Describe the Trigonometric Identities** 10

**Describe the sign of Trigonometric Functions in Different Quadrants** 12

**Describe the Trigonometric Functions of Allied Angles** 14

**Recall the Graph of various Trigonometric Functions and find their Domain and Range** 16

**Recall the Algebraic Sum of two or more angles of Trigonometric Functions** 17

**Recall the Algebraic difference of two or more angles of Trigonometric Functions** 19

**Find the Trigonometric ratios of multiple angles of 2A in terms of A** 23

**Formulate Sub-multiple angle A/2 in terms of A** 25

**Recall the multiple angles of 3A** 26

**Describe the Sum and Product Formulae of Trigonometric Ratios** 27

**Find the Height and Distance of various objects using Trigonometric ratios** 31

**Recall the Angle of Elevation and Angle of Depression** 32

**Recall Triangle and its Types**

In geometry, a triangle is a 3-sided polygon made up of three-line segments that intersect at three vertices. Triangles are one of the simplest and most common shapes in geometry, and they are used in a wide variety of mathematical and scientific applications.

There are several different ways to classify triangles based on their properties:

- Based on sides:
- Scalene Triangle: A triangle in which all three sides have different lengths.
- Isosceles Triangle: A triangle in which two sides have the same length, and the third side has a different length.
- Equilateral Triangle: A triangle in which all three sides have the same length.

- Based on angles:
- Acute Triangle: A triangle in which all three angles are acute (less than 90 °).
- Right Triangle: A triangle in which one angle is a right angle (exactly 90 °).
- Obtuse Triangle: A triangle in which one angle is obtuse (greater than 90 °).

- Based on both sides and angles:
- Right Isosceles Triangle: A triangle in which one angle is a right angle and the two sides adjacent to the right angle are of equal length.

For example, a triangle with sides of length 3, 4, and 5 is a right triangle because it satisfies the Pythagorean theorem. A triangle with sides of length 2, 2, and 3 is an isosceles triangle. An equilateral triangle has all sides of equal length, such as a triangle with sides of length 5, 5, and 5.

Knowing the types of triangles and their properties is important in various mathematical and scientific applications, such as in trigonometry, calculus, and physics.

**Describe the Properties of Triangles**

This Learning Outcome requires you to describe the properties of triangles. Triangles are a fundamental shape in geometry and they have various properties that are important to know for many mathematical applications. Here are some detailed notes on the properties of triangles:

- Types of triangles: There are three types of triangles based on their sides: scalene, isosceles, and equilateral. A scalene triangle has no equal sides, an isosceles triangle has two equal sides, and an equilateral triangle has all sides of equal length.

Example: In the triangle with sides of 3, 4, and 5 units, each side has a different length, making it a scalene triangle.

- Types of triangles: There are three types of triangles based on their angles: acute, right, and obtuse. An acute triangle has all angles less than 90 °, a right triangle has one angle equal to 90 °, and an obtuse triangle has one angle greater than 90 °.

Example: In the triangle with angles of 30, 60, and 90 °, the right angle is 90 °, making it a right triangle.

- Sum of angles: The sum of the angles of a triangle is always 180 °. This property is known as the Angle Sum Property of Triangles.

Example: In a triangle with angles of 60, 80, and 40 °, the sum of the angles is 180 °.

- Exterior angle: The exterior angle of a triangle is equal to the sum of the two interior angles not adjacent to it. This property is known as the Exterior Angle Property of Triangles.

Example: In the triangle with angles of 60, 80, and 40 °, the exterior angle adjacent to the 80-degree angle is 100 °, as it is equal to the sum of the two interior angles not adjacent to it (60 + 40).

- Pythagorean theorem: In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This property is known as the Pythagorean Theorem.

Example: In the right triangle with legs of length 3 and 4 units, the hypotenuse has a length of 5 units, as 3^{2} + 4^{2 }= 5^{2}

These are some of the key properties of triangles that are important to know. Understanding these properties can help you solve a variety of geometry problems, from finding the area of a triangle to using trigonometric ratios to solve for missing sides and angles. ** **

**Convert the Angle Measurement of ° into Radians and vice-versa**

This Learning Outcome requires you to convert between the angle measurement of ° and radians. The radian is the standard unit for measuring angles in mathematics, but ° are more commonly used in everyday life. Here are some detailed notes on how to convert between these two units:

- Conversion from ° to radians: To convert an angle measurement in ° to radians, you need to multiply the degree measure by π/180.

Example: To convert an angle of 45 ° to radians, we can use the formula: radians = (° × π) / 180 = (45 × π) / 180 = π/4. Therefore, 45 ° is equivalent to π/4 radians.

- Conversion from radians to °: To convert an angle measurement in radians to °, you need to multiply the radian measure by 180/π.

Example: To convert an angle of π/3 radians to °, we can use the formula: ° = (radians × 180) / π = (π/3 × 180) / π = 60. Therefore, π/3 radians is equivalent to 60 °.

- Understanding the relationship between ° and radians: There are 360 ° in a full circle, and 2π radians in a full circle. This means that one degree is equivalent to π/180 radians, and one radian is equivalent to 180/π °.

Example: A full circle has 360 ° or 2π radians. Therefore, one degree is equivalent to (2π/360) radians, which simplifies to π/180 radians. Similarly, one radian is equivalent to (360/2π) °, which simplifies to 180/π °.

- Converting between ° and radians in trigonometric functions: Trigonometric functions like sine, cosine, and tangent take input in radians, so you need to convert angle measurements in ° to radians before using them in these functions.

Example: To find the value of sin(30 °), we need to convert 30 ° to radians, which is π/6. Therefore, sin(30 °) = sin(π/6) = 0.5.

These are some of the key concepts and formulas you need to know in order to convert between the angle measurement of ° and radians. It is important to be comfortable with these conversions as they are used extensively in various areas of mathematics, physics, and engineering.

**Describe the Properties of Right Angle Triangles**

This Learning Outcome requires you to describe the properties of right angle triangles. A right angle triangle is a triangle that has one angle equal to 90 °. These triangles have a number of unique properties that are important to know for many mathematical applications. Here are some detailed notes on the properties of right angle triangles:

- Pythagorean theorem: In a right angle triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This property is known as the Pythagorean Theorem.

Example: In the right triangle with legs of length 3 and 4 units, the hypotenuse has a length of 5 units, as 3^{2} + 4^{2} = 5^{2}

- Trigonometric ratios: Right angle triangles are particularly useful for understanding trigonometric functions, which relate the lengths of the sides of a triangle to its angles. The three basic trigonometric ratios are sine, cosine, and tangent.

- Sine: The sine of an angle in a right angle triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
- Cosine: The cosine of an angle in a right angle triangle is the ratio of the length of the adjacent side to the length of the hypotenuse.
- Tangent: The tangent of an angle in a right angle triangle is the ratio of the length of the side opposite the angle to the length of the adjacent side.

Example: In the right triangle with a hypotenuse of length 5 and an adjacent side of length 3 units, the sine of the angle opposite the adjacent side is 4/5, as the opposite side has length 4 units. The cosine of the same angle is 3/5, and the tangent is 4/3.

- Special triangles: Right angle triangles with certain ratios of side lengths have special properties. These triangles are the 30-60-90 triangle and the 45-45-90 triangle.

- 30-60-90 triangle: In a 30-60-90 triangle, the side opposite the 30-degree angle has half the length of the hypotenuse, and the side opposite the 60-degree angle is the product of the hypotenuse and the square root of 3 divided by 2.
- 45-45-90 triangle: In a 45-45-90 triangle, the sides opposite the 45-degree angles are equal, and each has a length equal to the hypotenuse divided by the square root of 2.

Example: In the 30-60-90 triangle with a hypotenuse of length 6, the side opposite the 30-degree angle has length 3, and the side opposite the 60-degree angle has length 3 times the square root of 3 divided by 2, or approximately 2.6 units.

These are some of the key properties of right angle triangles that are important to know. Understanding these properties can help you solve a variety of geometry problems, from finding the length of a side to using trigonometric ratios to solve for missing sides and angles. ** **

**Describe the Six Trigonometric Functions**

This Learning Outcome requires you to describe the six trigonometric functions. Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of a triangle. The six trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent. Here are some detailed notes on each of these functions:

- Sine: The sine function is defined as the ratio of the length of the side opposite an angle in a right angle triangle to the length of the hypotenuse. The sine function is denoted by sin.

sin(θ) = opposite/hypotenuse

Example: In the right triangle with a hypotenuse of length 10 and an opposite side of length 6 units, the sine of the angle opposite the opposite side is 6/10, or 0.6.

- Cosine: The cosine function is defined as the ratio of the length of the adjacent side to an angle in a right angle triangle to the length of the hypotenuse. The cosine function is denoted by cos.

cos(θ) = adjacent/hypotenuse

Example: In the same right triangle as before, if the adjacent side has a length of 8 units, then the cosine of the angle adjacent to the adjacent side is 8/10, or 0.8.

- Tangent: The tangent function is defined as the ratio of the length of the side opposite an angle in a right angle triangle to the length of the adjacent side. The tangent function is denoted by tan.

tan(θ) = opposite/adjacent

Example: In the same right triangle as before, the tangent of the angle opposite the opposite side is 6/8, or 0.75.

- Cosecant: The cosecant function is defined as the reciprocal of the sine function.

csc(θ) = 1/sin(θ)

Example: In the same right triangle as before, the cosecant of the angle opposite the opposite side is 1/0.6, or approximately 1.67.

- Secant: The secant function is defined as the reciprocal of the cosine function.

sec(θ) = 1/cos(θ)

Example: In the same right triangle as before, the secant of the angle adjacent to the adjacent side is 1/0.8, or approximately 1.25.

- Cotangent: The cotangent function is defined as the reciprocal of the tangent function.

cot(θ) = 1/tan(θ)

Example: In the same right triangle as before, the cotangent of the angle opposite the opposite side is 1/0.75, or approximately 1.33.

These six trigonometric functions are important tools in trigonometry and have many applications in physics, engineering, and other fields. By understanding these functions, you can use trigonometry to solve a wide range of problems, from determining the height of a tree to calculating the trajectories of objects in motion. ** **

**Recall the Basic Trigonometric Ratios**

This Learning Outcome requires you to recall the basic trigonometric ratios. Trigonometric ratios are the ratios of the sides of a right triangle to its angles. The three basic trigonometric ratios are sine, cosine, and tangent. Here are some detailed notes on each of these ratios:

- Sine: The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. The sine ratio is denoted by sin.

sin(θ) = opposite/hypotenuse

Example: In the right triangle with a hypotenuse of length 10 and an opposite side of length 6 units, the sine of the angle opposite the opposite side is 6/10, or 0.6.

- Cosine: The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the angle to the length of the hypotenuse. The cosine ratio is denoted by cos.

cos(θ) = adjacent/hypotenuse

Example: In the same right triangle as before, if the adjacent side has a length of 8 units, then the cosine of the angle adjacent to the adjacent side is 8/10, or 0.8.

- Tangent: The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the adjacent side. The tangent ratio is denoted by tan.

tan(θ) = opposite/adjacent

Example: In the same right triangle as before, the tangent of the angle opposite the opposite side is 6/8, or 0.75.

These basic trigonometric ratios are used in a wide range of applications in mathematics, physics, engineering, and other fields. They can be used to solve problems related to angles and sides of a right triangle. By memorizing these basic ratios, you can quickly solve problems related to trigonometry. For example, if you know the length of two sides of a right triangle, you can use the basic trigonometric ratios to determine the length of the third side and the size of the angles.

**Describe the Trigonometric Identities**

This Learning Outcome requires you to describe the trigonometric identities. Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. These identities can be used to simplify trigonometric expressions and solve trigonometric equations. Here are some of the important trigonometric identities:

- Pythagorean identity: This is one of the most basic trigonometric identities, which relates the sine and cosine functions to each other. It is based on the Pythagorean theorem for a right triangle. According to this identity,

sin^{2}θ + cos^{2}θ = 1

Example: If the sine of an angle is 0.6, then the cosine of the same angle can be found as follows:

cos^{2}θ = 1 – sin^{2}θ = 1 – 0.6^{2} = 0.64

cos^{ }θ = sqrt(0.64) = 0.8

- Reciprocal identities: These identities involve the reciprocals of trigonometric functions. They are based on the definitions of trigonometric functions as ratios of sides of a right triangle. According to these identities,

sin^{2}θ = 1/cosec^{ }θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Example: If the sine of an angle is 0.6, then the cosecant of the same angle can be found as follows:

cosec θ= 1/sin θ = 1/0.6 = 1.67

- Quotient identity: This identity involves the quotient of the sine and cosine functions. According to this identity,

tanθ= sin θ/cosθ

Example: If the cosine of an angle is 0.8, and the sine of the same angle is 0.6, then the tangent of the angle can be found as follows:

tan θ = sin θ/cos θ = 0.6/0.8 = 0.75

- Co -function identity: This identity relates the sine and cosine functions of complementary angles. According to this identity,

sin(π/2 – θ) = cos (θ)

cos(π/2 – θ) = sin(θ)

Example: If the sine of an angle is 0.6, then the cosine of its complement (90 ° minus the angle) can be found as follows:

cos(90 – θ) = sin(θ) = 0.6

These are just a few of the many trigonometric identities that exist. By using these identities, you can simplify complicated trigonometric expressions and solve problems related to trigonometry.

**Describe the sign of Trigonometric Functions in Different Quadrants**

This Learning Outcome requires you to describe the sign of trigonometric functions in different quadrants. In trigonometry, the values of the trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) change sign depending on the quadrant in which the angle is located. This is important to understand when working with trigonometric functions and solving problems related to trigonometry. Here are the signs of the trigonometric functions in each of the four quadrants:

- In the first quadrant (0 to 90 °), all the trigonometric functions are positive. This is because in the first quadrant, both the x and y coordinates of the points on the unit circle are positive.

For example, if an angle of 30 ° is in the first quadrant, then:

sin(30) = ½

cos(30) = sqrt(3)/2

tan(30) = 1/sqrt(3)

- In the second quadrant (90 to 180 °), only the sine and cosecant functions are positive. This is because in the second quadrant, the x-coordinate of the points on the unit circle is negative, while the y-coordinate is positive.

For example, if an angle of 135 ° is in the second quadrant, then:

sin(135) = sqrt(2)/2

cos(135) = -sqrt(2)/2

tan(135) = -1

- In the third quadrant (180 to 270 °), only the tangent and cotangent functions are positive. This is because in the third quadrant, both the x and y coordinates of the points on the unit circle are negative.

For example, if an angle of 225 ° is in the third quadrant, then:

sin(225) = -sqrt(2)/2

cos(225) = -sqrt(2)/2

tan(225) = 1

- In the fourth quadrant (270 to 360 °), only the cosine and secant functions are positive. This is because in the fourth quadrant, the x-coordinate of the points on the unit circle is positive, while the y-coordinate is negative.

For example, if an angle of 315 ° is in the fourth quadrant, then:

sin(315) = -½

cos(315) = sqrt(3)/2

tan(315) = -1/sqrt(3)

It is important to remember these signs when working with trigonometric functions, as they help to identify the correct values of the functions in each quadrant.

**Describe the Trigonometric Functions of Allied Angles**

This Learning Outcome requires you to describe the trigonometric functions of allied angles. Allied angles are angles that differ by a multiple of 90 °. In trigonometry, the trigonometric functions of allied angles have special relationships that can be used to simplify trigonometric expressions and solve trigonometric equations. Here are the relationships for the trigonometric functions of allied angles:

- Sine and Cosine Functions:

- sin(90° + θ) = cos(θ)
- sin(90° – θ) = cos(θ)
- cos(90° + θ) = -sin(θ)
- cos(90° – θ) = sin(θ)

For example, suppose we want to find the value of sin(150°). Since 150° is not a special angle, we can use the relationship above to write:

sin(150°) = sin(90° + 60°) = cos(60°) = 1/2

- Tangent and Cotangent Functions:

- tan(90° + θ) = -cot(θ)
- tan(90° – θ) = cot(θ)
- cot(90° + θ) = -tan(θ)
- cot(90° – θ) = tan(θ)

For example, suppose we want to find the value of tan(240°). Since 240° is not a special angle, we can use the relationship above to write:

tan(240°) = tan(270° – 30°) = -cot(30°) = -1/√3

- Secant and Cosecant Functions:

- sec(90° + θ) = -csc(θ)
- sec(90° – θ) = csc(θ)
- csc(90° + θ) = -sec(θ)
- csc(90° – θ) = sec(θ)

For example, suppose we want to find the value of sec(120°). Since 120° is not a special angle, we can use the relationship above to write:

sec(120°) = sec(90° + 30°) = -csc(30°) = -2/√3

These relationships can be used to simplify trigonometric expressions, solve trigonometric equations, and find the values of trigonometric functions for angles that are not special angles. It is important to memorise these relationships and understand how they can be applied to different situations.

**Recall the Graph of various Trigonometric Functions and find their Domain and Range**

The graphs of various trigonometric functions along with their domains and ranges:

- Sine Function (sin x):

Graph: A periodic wave that oscillates between -1 and 1.

Domain: All real numbers, i.e., (-∞, ∞).

Range: [-1, 1].

- Cosine Function (cos x):

Graph: A periodic wave that oscillates between -1 and 1, but shifted by π/2 units to the right.

Domain: All real numbers, i.e., (-∞, ∞).

Range: [-1, 1].

- Tangent Function (tan x):

Graph: A periodic wave that has vertical asymptotes at odd multiples of π/2 and crosses the x-axis at even multiples of π/2.

Domain: All real numbers except odd multiples of π/2, i.e., (-∞, -π/2) U (-π/2, π/2) U (π/2, 3π/2) U (3π/2, ∞).

Range: All real numbers, i.e., (-∞, ∞).

- Cotangent Function (cot x):

Graph: A periodic wave that has vertical asymptotes at even multiples of π/2 and crosses the x-axis at odd multiples of π/2.

Domain: All real numbers except even multiples of π/2, i.e., (-∞, 0) U (0, π) U (π, 2π) U (2π, ∞).

Range: All real numbers, i.e., (-∞, ∞).

- Secant Function (sec x):

Graph: A periodic wave that oscillates between -1 and 1, but shifted by π/2 units to the left and has vertical asymptotes at odd multiples of π.

Domain: All real numbers except odd multiples of π, i.e., (-∞, -π) U (-π, 0) U (0, π) U (π, 2π) U (2π, ∞).

Range: [-1, -1] U [1, ∞) or (-∞, -1] U [-1, 1] U [1, ∞), depending on the convention.

- Cosecant Function (cosec x):

Graph: A periodic wave that oscillates between -1 and 1 and has vertical asymptotes at multiples of π.

Domain: All real numbers except multiples of π, i.e., (-∞, -π) U (-π, 0) U (0, π) U (π, 2π) U (2π, ∞).

Range: [-1, -1] U (-∞, -1] U [-1, 1] U [1, ∞), depending on the convention.

**Recall the Algebraic Sum of two or more angles of Trigonometric Functions**

This Learning Outcome requires you to recall the algebraic sum of two or more angles of trigonometric functions. The algebraic sum of two or more angles of trigonometric functions can be obtained by using the trigonometric identities that relate the sum of two angles to the products and differences of the trigonometric functions of the individual angles. The most commonly used trigonometric identities for the sum of two angles are:

- Sum of angles for sine function:

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

- Sum of angles for cosine function:

cos(a + b) = cos(a)cos(b) – sin(a)sin(b)

- Sum of angles for tangent function:

tan(a + b) = (tan(a) + tan(b))/(1 – tan(a)tan(b))

Using these identities, we can find the algebraic sum of two or more angles of trigonometric functions. For example:

- sin(π/4 + π/6)

Using the sum of angles for sine function:

sin(π/4 + π/6) = sin(π/4)cos(π/6) + cos(π/4)sin(π/6)

= (1/√2)(√3/2) + (1/√2)(1/2)

= (√6 + √2)/(2√2)

- cos(π/3 – π/6)

Using the sum of angles for cosine function:

cos(π/3 – π/6) = cos(π/3)cos(π/6) + sin(π/3)sin(π/6)

= (1/2)(√3/2) + (√3/2)(1/2)

= √3/2

- tan(π/4 + π/6)

Using the sum of angles for tangent function:

tan(π/4 + π/6) = (tan(π/4) + tan(π/6))/(1 – tan(π/4)tan(π/6))

= (1 + √3/3)/(1 – 1/3)

= 2 + √3

It is important to remember these trigonometric identities and how to use them to find the algebraic sum of two or more angles of trigonometric functions. This knowledge can be used to simplify trigonometric expressions, solve trigonometric equations, and find the values of trigonometric functions at specific angles.

**Recall the Algebraic difference of two or more angles of Trigonometric Functions**

This Learning Outcome requires you to recall the algebraic difference of two or more angles of trigonometric functions. The algebraic difference of two or more angles of trigonometric functions can be obtained by using the trigonometric identities that relate the difference of two angles to the products and sums of the trigonometric functions of the individual angles. The most commonly used trigonometric identities for the difference of two angles are:

- Difference of angles for sine function:

sin(a – b) = sin(a)cos(b) – cos(a)sin(b)

- Difference of angles for cosine function:

cos(a – b) = cos(a)cos(b) + sin(a)sin(b)

- Difference of angles for tangent function:

tan(a – b) = (tan(a) – tan(b))/(1 + tan(a)tan(b))

Using these identities, we can find the algebraic difference of two or more angles of trigonometric functions. For example:

- sin(π/3 – π/6)

Using the difference of angles for sine function:

sin(π/3 – π/6) = sin(π/3)cos(π/6) – cos(π/3)sin(π/6)

= (√3/2)(√3/2) – (1/2)(1/2)

= √3/2 – 1/4

- cos(5π/6 – π/6)

Using the difference of angles for cosine function:

cos(5π/6 – π/6) = cos(5π/6)cos(π/6) + sin(5π/6)sin(π/6)

= (-√3/2)(√3/2) + (-1/2)(1/2)

= -√3/4 – 1/4

- tan(3π/4 – π/4)

Using the difference of angles for tangent function:

tan(3π/4 – π/4) = (tan(3π/4) – tan(π/4))/(1 + tan(3π/4)tan(π/4))

= (-1 – 1)/(1 – (-1)(1))

= 0

It is important to remember these trigonometric identities and how to use them to find the algebraic difference of two or more angles of trigonometric functions. This knowledge can be used to simplify trigonometric expressions, solve trigonometric equations, and find the values of trigonometric functions at specific angles.

**Find the values of angles 15 degree, 75 degree, and 105 degree of Trigonometric Ratios using formulation of Sum and Difference of Angles**

Recall the formulas for the sum and difference of angles:

- sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
- cos(a + b) = cos(a)cos(b) – sin(a)sin(b)
- tan(a + b) = (tan(a) + tan(b))/(1 – tan(a)tan(b))
- sin(a – b) = sin(a)cos(b) – cos(a)sin(b)
- cos(a – b) = cos(a)cos(b) + sin(a)sin(b)
- tan(a – b) = (tan(a) – tan(b))/(1 + tan(a)tan(b))

Let’s first consider 15 °. We can express 15 ° as the sum or difference of angles for which we know the trigonometric ratios. For example, we can write:

15 ° = 45 ° – 30 °

Using the difference of angles formula for sine, we can find the value of sin(15 °):

sin(15 °) = sin(45 ° – 30 °) = sin(45 °)cos(30 °) – cos(45 °)sin(30 °)

= (√2/2)(√3/2) – (√2/2)(1/2)

= (√6 – √2)/4

Similarly, we can use the formulas for sum and difference of angles to find the values of other trigonometric ratios for 15 °.

For 75 °, we can write:

75 ° = 45 ° + 30 °

Using the sum of angles formula for sine, we can find the value of sin(75 °):

sin(75 °) = sin(45 ° + 30 °) = sin(45 °)cos(30 °) + cos(45 °)sin(30 °)

= (√2/2)(√3/2) + (√2/2)(1/2)

= (√6 + √2)/4

For 105 °, we can write:

105 ° = 135 ° – 30 °

Using the difference of angles formula for cosine, we can find the value of cos(105 °):

cos(105 °) = cos(135 ° – 30 °) = cos(135 °)cos(30 °) + sin(135 °)sin(30 °)

= (-√2/2)(√3/2) + (-√2/2)(1/2)

= (-√6 – √2)/4

Using the formulas for sum and difference of angles to find the values of trigonometric ratios for specific angles can be a useful skill when solving trigonometric equations or evaluating trigonometric functions at non-standard angles.

**Find the Trigonometric ratios of multiple angles of 2A in terms of A**

In trigonometry, the trigonometric ratios of multiple angles of 2A can be found in terms of A using the double-angle formulas. These formulas allow us to express trigonometric functions of an angle in terms of trigonometric functions of half the angle. The formulas are as follows:

- Sin(2A) = 2sin(A)cos(A)
- Cos(2A) = cos²(A) – sin²(A) = 2cos²(A) – 1 = 1 – 2sin²(A)
- Tan(2A) = 2tan(A) / (1 – tan²(A))

Using these formulas, we can find the trigonometric ratios of 2A in terms of A. Let’s take some examples to understand the application of these formulas.

Example 1: Find sin(30°).

We know that sin(30°) = 0.5. Now we can find sin(2*15°) using the double-angle formula for sin:

sin(2A) = 2sin(A)cos(A)

sin(2*15°) = 2sin(15°)cos(15°)

We can use the half-angle formulas to find sin(15°) and cos(15°):

sin(15°) = √[(1 – cos(30°)) / 2] = √[(1 – √3/2) / 2]

cos(15°) = √[(1 + cos(30°)) / 2] = √[(1 + √3/2) / 2]

Substituting these values, we get:

sin(2*15°) = 2 * √[(1 – √3/2) / 2] * √[(1 + √3/2) / 2] = √[(3 – √3) / 4]

Therefore, sin(30°) = sin(2*15°) / 2 = √[(3 – √3) / 8].

Example 2: Find tan(60°).

We know that tan(60°) = √3. Now we can find tan(2*30°) using the double-angle formula for tan:

tan(2A) = 2tan(A) / (1 – tan²(A))

tan(2*30°) = 2tan(30°) / (1 – tan²(30°))

We can use the half-angle formula to find tan(30°):

tan(30°) = sin(30°) / cos(30°) = √3/3

Substituting this value, we get:

tan(2*30°) = 2 * (√3/3) / (1 – (√3/3)²) = √3

Therefore, tan(60°) = tan(2*30°) = √3.

**Formulate Sub-multiple angle A/2 in terms of A**

In trigonometry, sub-multiple angles are angles that are fractions of the larger angle, usually half or a third. They are important in finding the values of trigonometric functions for certain angles.

One important sub-multiple angle is half-angle, or A/2, which can be used to simplify trigonometric expressions involving even powers of trigonometric functions.

To formulate sub-multiple angle A/2 in terms of A, we can use the following trigonometric identities:

sin(A/2) = ±√[(1 – cos A)/2]

cos(A/2) = ±√[(1 + cos A)/2]

tan(A/2) = ±√[(1 – cos A)/(1 + cos A)]

The choice of sign depends on the quadrant in which angle A lies.

For example, if A = 60 °, we can find the values of sin(30) and cos(30) using the half-angle formulas as follows:

cos A = cos 60 = ½

cos(30) = cos(60/2) = ±√[(1 + cos 60)/2] = ±√[(1 + 1/2)/2] = ±√(3/4)

Since 30 ° is in the first quadrant, we choose the positive sign, so cos(30) = √(3)/2.

Similarly, sin(30) = sin(60/2) = ±√[(1 – cos 60)/2] = ±√[(1 – 1/2)/2] = ±√(1/4)

Again, since 30 ° is in the first quadrant, we choose the positive sign, so sin(30) = 1/2.

Therefore, the values of sin(30) and cos(30) are 1/2 and √(3)/2, respectively, which are the familiar values for the sine and cosine of 30 °.

**Recall the multiple angles of 3A**

In trigonometry, the multiple angles of 3A are the angles that are three times the size of the angle A. These angles are important in simplifying and solving trigonometric expressions, as they allow us to express trigonometric functions in terms of a single angle, which can be easier to work with.

To recall the multiple angles of 3A, we can use the following trigonometric identities:

sin(3A) = 3 sinA – 4sin³A

cos(3A) = 4 cos³A – 3cosA

tan(3A) = (3 tanA – tan³A)/(1 – 3tan²A)

For example, suppose we want to find the value of sin(60°). We can use the multiple angle formula for sin(3A) as follows:

sin(3A) = 3 sinA – 4 sin³A

Let A = 20°, so 3A = 60°. Then, we have:

sin (60°) = sin (3A) = 3 sin (20°) – 4 sin³ (20°)

To find the values of sin(20°), we can use the half-angle formula:

sin (20°) = 2 sin (10°) cos(10°)

We know that sin(10°) = 0.1736 and cos(10°) = 0.9848, so:

sin(20°) = 2(0.1736)(0.9848) = 0.3406

Substituting this value back into the original equation, we have:

sin(60°) = 3sin(20°) – 4sin³(20°) = 3(0.3406) – 4(0.3406)³ = 0.8660

Therefore, sin(60°) = 0.8660, which is the expected value for the sine of 60 °.

**Describe the Sum and Product Formulae of Trigonometric Ratios**

The Sum and Product Formulae are used to calculate the trigonometric ratios of the sum and difference of two or more angles. These formulas are widely used in calculus, engineering, and physics.

- Sum Formula:

The sum formulae give the trigonometric ratios of the sum of two angles. They are as follows:

- sin(A+B) = sin(A)cos(B) + cos(A)sin(B)
- cos(A+B) = cos(A)cos(B) – sin(A)sin(B)
- tan(A+B) = (tan(A) + tan(B)) / (1 – tan(A)tan(B))
- cot(A+B) = (cot(A)cot(B) – 1) / (cot(B) + cot(A))
- sec(A+B) = (sec(A)sec(B)) / (sec(A)cos(B) + cos(A)sec(B))
- csc(A+B) = (csc(A)csc(B)) / (csc(B)cos(A) + cos(B)csc(A))

- Product Formulae:

The product formulae give the trigonometric ratios of the product of two angles. They are as follows:

- sin(A)sin(B) = (cos(A-B) – cos(A+B)) / 2
- cos(A)cos(B) = (cos(A-B) + cos(A+B)) / 2
- sin(A)cos(B) = (sin(A+B) + sin(A-B)) / 2
- tan(A)tan(B) = (1 – tan(A+B)) / (1 + tan(A)tan(B))
- cot(A)cot(B) = (cot(A+B) + 1) / (cot(A) + cot(B))

Example:

Let’s use the sum formula to find the value of sin(45° + 60°).

sin(45° + 60°) = sin(45°)cos(60°) + cos(45°)sin(60°)

Using the values of sin(45°) = cos(45°) = √2 / 2 and cos(60°) = 1 / 2 and sin(60°) = √3 / 2, we get:

sin(45° + 60°) = (√2 / 2) × (1 / 2) + (√2 / 2) × (√3 / 2)

= (√2 / 4) + (√6 / 4)

= (√2 + √6) / 4

Hence, sin(45° + 60°) = (√2 + √6) / 4.

**Write the product of Trigonometric Ratios in the form of Sum/Difference of Trigonometric ratios and vice-versa**

Trigonometric functions can be expressed as products of trigonometric ratios in some cases, and it may be necessary to express these products as sums or differences of trigonometric functions. This is particularly useful in simplifying complex expressions involving trigonometric functions.

Sum and Difference Formulas, which were discussed in ALO:

- sin(a) sin(b) = (cos(a-b) – cos(a+b))/2
- cos(a) cos(b) = (cos(a-b) + cos(a+b))/2
- sin(a) cos(b) = (sin(a+b) + sin(a-b))/2
- cos(a) sin(b) = (sin(a+b) – sin(a-b))/2

For example, if we have to simplify the following expression, using product-to-sum identity: sin 20 sin 40 sin 80

We can write the given product as follows:

sin 20 sin 40 sin 80 = (1/2) [(cos 20 – cos 60)(cos 40 – cos 40)(cos 80 – cos 0)]

Using the formula sin A = cos (90 – A) for angle transformations, we can simplify this expression as:

= (1/8) [(cos 20 – cos 60)(1 – 2sin^{2} 40)(cos 80 – 1)]

Expanding this expression and simplifying further gives:

= (1/16) [cos 20 cos 80 – cos 20 – cos 60 cos 80 + cos 60 – cos 20 cos 80 + cos 80]

= (1/16) [cos 60 – cos 20 – cos 80]

Hence, the product of sin 20 sin 40 sin 80 can be expressed as the difference of three cosine functions.

Vice versa, if we have a trigonometric function in the form of a sum or difference, we can use the Sum and Difference Formulas to express it as a product of trigonometric ratios.

For example, if we want to express the following function as a product of trigonometric functions:

cos 30 + cos 50

Using the formula cos A + cos B = 2 cos [(A+B)/2] cos [(A-B)/2], we can write the given function as:

cos 30 + cos 50 = 2 cos [(30+50)/2] cos [(50-30)/2]

Simplifying this expression gives:

cos 30 + cos 50 = √3 cos 10

Hence, the sum of two cosine functions can be expressed as a product of a cosine function and a constant.

**Find the Height and Distance of various objects using Trigonometric ratios**

Trigonometry is widely used to find the heights and distances of objects that are difficult to measure directly. In order to find the height and distance of objects, we use the trigonometric ratios of angles.

The trigonometric ratios used to find the height and distance of objects are sine, cosine, and tangent.

Let’s consider an example:

Suppose you are standing at point A and want to find the height of a tree (point B) and the distance from your point of view to the tree. You can use the following steps to find the height and distance of the tree.

Step 1: Stand at point A and mark a point C on the ground at some distance from the tree. Measure the distance AC.

Step 2: Measure the angle of elevation, which is the angle between the horizontal and the line of sight from your eye to the top of the tree (point B). Let’s assume that the angle of elevation is 30 °.

Step 3: Using the trigonometric ratios, we can calculate the height of the tree as follows:

height of tree = AC x tan(30)

Step 4: Using the trigonometric ratios, we can also calculate the distance from your point of view to the tree as follows:

distance from point A to the tree = AC / cos(30)

In this example, if AC is 30 meters, then the height of the tree would be approximately 17.3 meters, and the distance from point A to the tree would be approximately 34.6 meters.

Trigonometric ratios can also be used to find the height and distance of other objects, such as buildings, towers, and mountains, as long as you have the angle of elevation and the distance from your point of view to the object.

**Recall the Angle of Elevation and Angle of Depression**

In trigonometry, the angle of elevation and angle of depression are two important terms used to describe the angle between a horizontal line and a line of sight to an object.

The angle of elevation is the angle that the line of sight makes with the horizontal when an observer is looking up at an object. For example, suppose an observer is standing on the ground and looking up at a bird sitting on a tree. The angle that the observer’s line of sight makes with the horizontal is the angle of elevation.

On the other hand, the angle of depression is the angle that the line of sight makes with the horizontal when an observer is looking down at an object. For example, suppose an observer is standing on the top of a hill and looking down at a boat floating in a lake. The angle that the observer’s line of sight makes with the horizontal is the angle of depression.

Both the angle of elevation and angle of depression are measured in degrees or radians. They are important in solving various real-life problems involving distances, heights, and angles.

For example, if the height of an object and the angle of elevation from a particular point are known, then the distance of the object from the point can be calculated using trigonometry. Similarly, if the distance of an object and the angle of depression from a particular point are known, then the height of the object can be calculated using trigonometry.

In summary, the angle of elevation and angle of depression are important concepts in trigonometry that are used to solve problems related to distances, heights, and angles in real-life situations.