Trigonometric Ratios of Compound Angles

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Before going to the article, let’s just know something about trigonometry and its usage in mathematics as well as physics.

Trigonometry is a division of mathematics, which is about the angles and their measurement. 

The functions under trigonometry consist of compound angles, trigonometric ratios, and multiple angles. These functions are applied to find out accurate results. 

We are going to discuss trigonometric ratios of compound angles in this content along with the explained questions and answers.

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An angle can be defined as a ray having its rotation around a startup point. Here, the point where the rotation starts is the initial, whereas the point where it stops is the final one. 

When you rotate the ray in an anticlockwise direction, it will make a positive angle. However, rotating the ray in the clockwise direction will make it negative. You can measure the angle either in degree or radians.


What are Compound Angles?

A compound angle is defined as the algebraic sum of two or more angles.

The compound angles do represent numerous trigonometric identities.

List of Trigonometric Ratios of Compound Angles

  1. cos A cos B – sin A cos B = cos (A + B)

  2. cos A cos B + sin A cos B = cos (A – B)

  3. sin A cos B + cos A sin B = sin (A + B)

  4. sin A cos B – cos A sin B = sin (A – B) 

  5. sin2 A – sin2 B = sin (A + B) sin (A – B) = cos2 B – cos2 A

  6. cos2 A – sin2 A – sin2 B = cos (A + B) cos (A – B) = cos2 B – sin2 A

  7. [tan A + tan B] / [1 – tan A tan B] = tan (A + B) 

  8. [tan A – tan B] / [1 + tan A tan B] = tan (A – B) 


For The Sum or Difference of Sines and Cosines

  1. sin (A+B) + sin (A-B) = 2 sin A cos B 

  2. sin (A+B) – sin (A-B) = 2 cos A sin B

  3. cos (A+B) + cos (A-B) = 2 cos A cos B

  4. cos (A-B) – cos (A+B) = 2 sin A sin B


For The Multiple and Sub-multiple Angles

  1. 2 sin A cos A = 2 tan A  /(1 + tan2 A) = sin 2A 

  2. cos 2A – sin2A = 2cos2A - 1 = 1 - 2 sin2A = cos 2A 

  3. 4 cos 3A – 3 cos A = cos 3A

  4. 3 sin A – 4 sin 3A = sin 3A

  5. 2 tan A / (1 - tan2A) = tan 2A 

  6. (3 tan A - tan3A) / (1 - tan2A) = tan 3A 


Trigonometric Ratios of Multiples of an Angle

  1. sin⁡2A = sin⁡(A+A) = sin⁡A.cos⁡A + cos⁡A.sin⁡A = 2sin⁡A.cos⁡A

  2. sin⁡2A = 2sin⁡A.cos⁡A = \[\frac{2sin⁡A.cos⁡A}{1}\] = \[\frac{2sin⁡A.cos⁡A}{cos^2A + sin^2A}\] = \[\frac{\frac{2sin⁡A.cos⁡A}{A}}{\frac{A}{A} + \frac{A}{A}}\] = \[\frac{2tanA}{1 + tan^2A}\]

  3. sin⁡2A = \[\frac{2tanA}{1 + tan^2A}\] = \[\frac{\frac{2}{cos cosA}}{1 + \frac{1}{A}}\] = \[\frac{2}{cotcotA}\] × \[\frac{A}{1 + cot^2A}\] = \[\frac{2cotA}{1+cot^2A}\]

  4. cos⁡2A = cos⁡(A+A) = cos⁡A.cos⁡A - sin⁡A.sin⁡A = cos2⁡A - sin2⁡A

  5. cos⁡2A = cos2⁡A - sin2⁡A = 1-sin2⁡A - sin2⁡A = 1-2sin2⁡A

  6. cos 2A = A - A = A - (1 - A) = 2A -1

  7. cos 2A = A - A = \[\frac{A - A}{1}\] = \[\frac{A - A}{A + A}\] = \[\frac{\frac{A}{A} - \frac{A}{A}}{\frac{A}{ω^2_qA} + \frac{A}{ω^2_lA}}\] = \[\frac{1-tan^2A}{1+tan^2A}\]

  8. cos⁡2A = \[\frac{1-tan^2A}{1+tan^2A}\] = \[\frac{1-\frac{1}{A}}{1+\frac{1}{A}}\] = \[\frac{cot^2A-1}{cot^2A+1}\]

  9. tan⁡2A = tan⁡(A+B) = \[\frac{tanA + tanA}{1-tanA.tanA}\] = \[\frac{2tanA}{1-tan^2A}\]

  10.  tan⁡2A = \[\frac{2tanA}{1-tan^2A}\] = \[\frac{\frac{2}{cotcotA}}{1-\frac{1}{A}}\] = \[\frac{2}{cotcotA}\] × \[\frac{A}{cot^2A-1}\]

  11.  cot⁡2A=cot⁡(A+A)= \[\frac{cotA.cotA-1}{cotA + cotA}\] = \[\frac{cot^2⁡A-1}{2cot⁡A}\]

  12.  cot⁡2A = \[\frac{cot^2⁡A-1}{2cot⁡A}\] = \[\frac{\frac{1}{A}-1}{\frac{2}{\text{tan tanA}}}\] = \[\frac{1-tan^2A}{A}\] × \[\frac{1-tan^2A}{2tanA}\] = \[\frac{1-tan^2⁡A}{2tanA}\]


Problems Using Compound Angle Formulae

These are some examples regarding compound angle formulae, These are:

1. Assume that A and B are positive acute angles, where sin A = 1 / √10 and sin B = 1 / √5. Calculate A + B.

Ans: Let’s find cos A and cos B

We know that 

sin2 A + cos2 A = 1

⇒ cos A = √ (1 - sin2 A)

⇒ cos A = √ (1 – 1 / 10)

Similarly, cos B = √ (1 – 1 / 10)

By putting the sin A and sin B value in the above equation, we get:

According to the formula,

sin (A + B) = sin A cos B + cos A sin B

= [1 / √10] [√ (1 − 1 / 5)] + [1 / √5] [√ (1 − 1 / 10)]

= [1 / √10] [√ (4 / 5)] + [1 / √5] [√ (9 / 10)]

= [1 / √50] * (2 + 3)

= 5 / √50 

= [5 / 5] (1 / √2)

= 1 / √ 2

⇒ sin (A + B) = 1 / √ 2

⇒ A + B = sin-1 (1 / √ 2)

⇒ A + B = π / 4

2. The value of x = (1 + tan A) (1 − tan B) and A − B = π / 4. Then, what will be the value of (x + 1)x+1?

Ans:

According to the question

A − B = π / 4 

Let’s multiply tan in both sides;

⇒ tan (A − B) = tan π / 4

By putting the tan (A – B), we get;

⇒ [tan A − tan B] / [1 + tan A * tan B] = 1

⇒ tan A − tan B − tan A * tan B = 1

Let’s add 1 in both sides of the equation

⇒ {tan A − tan B −tan A * tan B + 1} = 1 + 1

We know that, (1 + tan A) (1 − tan B) = {tan A − tan B −tan A * tan B + 1}

So, (1 + tan A) (1 − tan B) = 2

⇒ x = 2

Putting x = 2 in (x + 1)x+1 

= (2 + 1)2+1 

= (3)

= 27

FAQ (Frequently Asked Questions)

Q1. Define Trigonometry and Write its application.

Ans. Trigonometry is the division of geometry widely used for the measurement of sides of a triangle. 

It is quite easy to solve the problems using trigonometry along with measurements of the triangles. 

Application of trigonometry is very popular among the navigators, scientists, meteorologists, and engineers, as it serves numerous purposes.

Q2. What do you mean by ‘Zero Angle’?

Ans. A zero angle is meant by an angle having a degree of zero. The process of visualization of zero angles is quite hard.

Take an example of two lines that create an angle greater than zero degrees. If both of the lines pass through each other without any inclination, then zero angles are possible.

Q3. What are the 3 Basic Trigonometric Ratios?

Ans. The names of the three basic trigonometric ratios are ‘sine’, ‘cosine’, and ‘tangent’.

In a right-angled triangle, we can find the sine, or cosine, or tangent for both of the non- 90° angles. If ABC is a right-angled triangle, we can write the expression for angleA using sine, cosine, and tangent.

Q4. What do you mean by the Double Angle?

Ans. The double angle is a concept which is related to the three common trigonometric ratios such as, 

  1. sine (sin)

  2. cosine (cos)

  3. tangent (tan)

The sole purpose of these ratios is to express the connection between the sides of a right-angle triangle to certain angles in that triangle.