Compound Angles
Values of some angles are already known to us through the value table i.e. 0°, 30°, 45°, 60° and 90°. But, there are many more angles whose values are not directly indicated by the value table. And value of those angles can be obtained by adding or subtracting the values of two known angles. Therefore, the angles which can be presented as the sum or difference of two known angles [ 0°, 30°, 45°, 60° or 90°] can be termed as compound angle. Eg: 15° = 45° -30°, 105° = 60° +45°.






Trigonometric Functions for Compound Angles:


Trigonometry is a branch of Mathematics that deals with the measurement and relations of angles and sides of a triangle.
For the Lower Secondary level education, compound angles in trigonometry should be better understood than memorized. We all know that we have six important functions in trigonometry viz. sin(sine), cos(cosine), tan(tangent), cot(cotangent), sec(secant) and cosec(cosecant). But, for the compound angles formula, we need to understand only four functions in this level i.e. sin, cos, tan and cot.

 Now, let us discuss about basic definition of angles:

Angles:

Basically, angle is the figure formed by joining two rays, sharing a common endpoint. Angles can range anywhere form 0° to 360° in Mathematics. 

Relation between degree and radians measures:
These terms are always somehow headache for the pupils if they have no idea about them. While in examination, they might loose their marks as well if they have no clear understanding.
When we measure an angle in term of x° , we are measuring in terms of degrees and when we measure in Î², we are measuring in radians.

The relation between radians and degree measures can be expressed as:
  • Radian measure = Pi/180* Degree measure
  • Degree measure = 180/Pi * Radian measure

Compound Angles:

As already stated in the introduction part, compound angles are the angles which can be expressed in the form of sum or difference of two angles [ 0°, 30°, 45°, 60° or 90°]. Values of these angles can be obtained by using the trigonometric identities. Here, we may overcome with the angles which can be in the form of (A+B) or (A-B). Depending on the situation, we have the formulas. The compound angles formulas are stated below:

1. sin(A+B) = sinA.cosB + cosA.sinB

2. sin(A-B) = sinA.cosB - cosA.sinB

3. cos(A+B) = cosA.cosB - sinA.sinB

4. cos(A-B) = cosA.cosB + sinA.sinB

We will discuss the derivation of these formulas later on in another posts. Now, lets prove the formulas of tan and cot using the compound angle formulas of sin and cos.
We know, 
tan = sin/ cos
cot = cos/ sin

5. tan(A+B) =  sin(A+B) / cos(A+B)

Therefore, tan(A+B) = (tanA + tanB) / (1 -tanA.tanB)

 6. tan(A+B) =  sin(A-B) / cos(A-B)

Therefore, tan(A-B) = (tanA-tanB) /(1-tanA.tanB)

 7. cot(A+B) = cos(A+B) / sin(A+B)
Therefore, cot(A+B) = (cotA.cotB -1) / (cotB + cotA)

8. cot(A-B) = cos(A-B) / sin(A-B)
Therefore, cot(A-B) = (cotA.cotB +1) / (cotB - cotA)
Now, that we know all the basic formulas of compound angles, let us understand an easy way to quickly save them in our memory.

Think that sin is opposite to cos and tan is opposite to cot.

Think sin and tan both are positive while cos and cot are negative.

This is our first step to understand the sign.

In formula for sine, the sign in the formula remains the same used as the argument. For eg: for sin(a+b), the sign in the middle in the formula remain + and for sin(a-b), the sign in the middle in the formula remain -. And the opposite is for cos.

For tan, the first sign is positive and the second one is negative. And its just opposite for cot.

This is how I remembered all those eight formulas.

Now, we have some more formulas, when we have both angles i.e.(A+B) same viz. A=B, our argument become (2A).

Now, formulas for double angles of (2A), we have:

9. sin(A+A) = sinA.cosA + cosA.sinA
    or, sin(2A) = sinA.cosA + sinA.cosA
Therfore, sin(2A) = 2sinA.cosA

 10. cos(A+A) = cosA.cosA - sinA.sinA
    or, cos(2A) = cos^2 A -  sin^2 A
sherfore, cos(2A) = cos^2 A -  sin^2 A

 Similarly,
 11. tan(2A) = (2tanA) / (1- tan^2 A)

 12. cot(2A) = (cot^2 A -1) / 2cotA

And, 
we have some more formulas that we need to understand. I am attaching the photos of all the formulas of compound angles together.






I hope that we have been clear about the formulas of compound angles. We will meet in another blog.
If you want to see some solved questions about Trigonometric Ratios of Compound Angles. 


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