Trigonometric Ratios of Compound Angles - Practice Questions & MCQ

Trigonometric Ratio for Compound Angles (Part 1)

The sum or difference of two or more angles is called a compound angle. If A, B and C are any angle then A + B, A - B, A + B + C, A + B - C etc all are compound angles.

1. cos (α - β) = cos α cos β + sin α sin β

2. cos (α+ β) = cos α cos β - sin α sin β

3. sin (α - β) = sin α cos β - cos α sin β 

4. sin (α + β) = sin α cos β + cos α sin β 

Proof:

1. cos (α - β) 

Let’s consider two points on the unit circle. Point P is at an angle α from the positive x-axis with coordinates (cos α, sin α) and point Q is at an angle of β from the positive x-axis with coordinates (cos β, sin β). Note the measure of angle POQ is α − β. Label two more points: A at an angle of (α − β) from the positive x-axis with coordinates (cos(α − β), sin(α − β)); and point B with coordinates (1, 0). Triangle POQ is a rotation of triangle AOB and thus the distance from P to Q is the same as the distance from A to B. 

We can find the distance from P to Q using the distance formula.

Similarly, using the distance formula we can find the distance from A to B.

Because the two distances are the same, we set them equal to each other and simplify

Finally we subtract 2 from both sides and divide both sides by −2

2. cos (α + β) 

3. Sine Compound Angle Formulae

Trigonometric Ratio for Compound Angles (Part 2)

Proof:

Finding the sum of two angles formula for tangent involves taking quotient of the sum formulas for sine and cosine and simplifying,

[Divide the numerator and denominator by cosα cos β]

For the difference of tangent, put - β in place of β in the above result

Cot Formulae

For the difference of cotangent, put - β in place of β in the above result

Trigonometric Ratio for Compound Angles (Part 3)

Proof Cotangent of the Sum and  Difference of two Angles

For the difference of cotangent, put - β in place of β in the above derivation.

Trigonometric Ratio for Compound Angles (Some more Results)

Proof:

1.

2.

3.