Compound Angle Formulae

In this page, we will will learn about the Compound Angle Formulae used in Trigonometry. These are really important because they open up so many other formulae in trigonometry. In particular, we can derive the double angle formula. There are 6 formula which are written in a shortened form in the IB formula booklet. You need to be very careful with positive/negative signs.


Key Concepts

On this page, you should learn about the double angle identities for sine and cosine

\(\large\sin(A+B)\equiv\sin A\cos B+ \cos A \sin B\\ \large\sin(A-B)\equiv\sin A\cos B- \cos A \sin B\)

\(\large\cos(A+B)\equiv\cos A\cos B- \sin A \sin B\\ \large\cos(A-B)\equiv\cos A\cos B+ \sin A \sin B\\\)

\(\large \tan(A+B)≡\frac{\tan A +\tan B}{1-\tan A\tan B}\\ \large \tan(A-B)≡\frac{\tan A -\tan B}{1+\tan A\tan B}\)

Summary

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Test Yourself

This quiz is about the Compound Angle formulae for sin(A+B) , cos(A+B) and tan(A+B)

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Exam-style Questions

Question 1

If \(\large \sin A=\frac{4}{5}\) , where \(\large 0\le A\le\frac{\pi}{2}\)

and \(\large \cos B=-\frac{12}{13}\) , where \(\large \pi \le B\le\frac{3\pi}{2}\)

work out \(\large \cos(B-A)\)

Hint

Full Solution

 

Question 2

If \(\large \sin(x+30°)=2\cos(x+60°)\), then show that \(\large \tan x=\frac{\sqrt{3}}{9}\)

Hint

Full Solution

 

Question 3

a) By writing 15° as 45° - 30° , find the value of sin15°

b) Hence, show that the area of this triangle \(\large =4(\sqrt{3}-1)\)

Hint

Full Solution

 

Question 4

Prove that

\(\large \frac{\sin(A+B)+\sin(A-B)}{\cos(A+B)+\cos(A-B)}=\tan A\)

Hint

Full Solution

 

Question 5

Prove that \(\large \tan 3x\equiv \frac{3\tan x-\tan^3x}{1-3\tan^2x}\)

Hint

Full Solution

 

Question 6

a) Write \(\large \cos4x\) in terms of \(\large\cos x\)

b) Hence, solve \(\large 8\cos^4x-8\cos^2x+1=\sin4x\) , for \(\large 0\le x\le\pi\)

Hint

Full Solution

 

MY PROGRESS

How much of Compound Angle Formulae have you understood?