In trigonometry, we already know the values of some angles through the value table, i.e. 0°, 30°, 45°, 60° and 90°. But, there are many more angles whose values that the value table does not directly indicate.
We can find the value of those angles by adding or subtracting the values of two known angles. Therefore, the angles that can be represented as the sum or the difference of two known angles [ 0°, 30°, 45°, 60° or 90°] are termed compound angles. 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 the basic definition of angles:
Angle: Basically, an angle is the figure formed by joining two rays, sharing a common endpoint. Angles can range anywhere from 0° to 360° in Mathematics.
Relation between Degree and Radians
These terms are always somehow headache for the pupils if they have no idea about them. While in an examination, they might lose their marks as well if they have no clear understanding. When we measure an angle in terms of x°, we are measuring in terms of degrees and when we measure in β, 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 in Trigonometry
As already stated in the introduction part, compound angles are the angles that can be expressed in the form of the sum or the 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 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 other posts. Now, let us prove the formulas of tan and cot using the compound angle formulas of sin and cos. We know, tan = sin/ cos and 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)
How to memorize the compound angle formula?
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 the formula for sine, the sign in the formula remains the same as the argument. For eg: for sin(a+b), the sign in the middle of the formula remains +, and for sin(a-b), the sign in the middle of the formula remains -. And the opposite is for cos.
For tan, the first sign is positive and the second one is negative. And it is just the opposite for cot.