A brief note on Sum and Difference Formulae . Read it before a test. Try each question at the end before you open its answer.
Sums and differences cos ( x + y ) = cos x cos y − sin x sin y \cos(x + y) = \cos x \cos y - \sin x \sin y cos ( x + y ) = cos x cos y − sin x sin y cos ( x − y ) = cos x cos y + sin x sin y \cos(x - y) = \cos x \cos y + \sin x \sin y cos ( x − y ) = cos x cos y + sin x sin y sin ( x ± y ) = sin x cos y ± cos x sin y \sin(x \pm y) = \sin x \cos y \pm \cos x \sin y sin ( x ± y ) = sin x cos y ± cos x sin y tan ( x ± y ) = tan x ± tan y 1 ∓ tan x tan y \tan(x \pm y) = \dfrac{\tan x \pm \tan y}{1 \mp \tan x \tan y} tan ( x ± y ) = 1 ∓ tan x tan y tan x ± tan y Allied angles With π \pi π or 2 π 2\pi 2 π the function stays; with π 2 \dfrac{\pi}{2} 2 π or 3 π 2 \dfrac{3\pi}{2} 2 3 π , sine and cosine swap. The sign is the sign of the first function in the quadrant where the angle lands, for a small acute x x x . For example, cos ( π 2 + x ) = − sin x \cos\left(\dfrac{\pi}{2} + x\right) = -\sin x cos ( 2 π + x ) = − sin x and sin ( π + x ) = − sin x \sin(\pi + x) = -\sin x sin ( π + x ) = − sin x . Double, half and triple angles sin 2 x = 2 sin x cos x = 2 tan x 1 + tan 2 x \sin 2x = 2\sin x \cos x = \dfrac{2\tan x}{1 + \tan^2 x} sin 2 x = 2 sin x cos x = 1 + tan 2 x 2 tan x cos 2 x = cos 2 x − sin 2 x = 2 cos 2 x − 1 = 1 − 2 sin 2 x \cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x cos 2 x = cos 2 x − sin 2 x = 2 cos 2 x − 1 = 1 − 2 sin 2 x tan 2 x = 2 tan x 1 − tan 2 x \tan 2x = \dfrac{2\tan x}{1 - \tan^2 x} tan 2 x = 1 − tan 2 x 2 tan x sin x 2 = ± 1 − cos x 2 \sin \dfrac{x}{2} = \pm\sqrt{\dfrac{1 - \cos x}{2}} sin 2 x = ± 2 1 − cos x , cos x 2 = ± 1 + cos x 2 \cos \dfrac{x}{2} = \pm\sqrt{\dfrac{1 + \cos x}{2}} cos 2 x = ± 2 1 + cos x , with the sign from the quadrant of x 2 \dfrac{x}{2} 2 x sin 3 x = 3 sin x − 4 sin 3 x \sin 3x = 3\sin x - 4\sin^3 x sin 3 x = 3 sin x − 4 sin 3 x and cos 3 x = 4 cos 3 x − 3 cos x \cos 3x = 4\cos^3 x - 3\cos x cos 3 x = 4 cos 3 x − 3 cos x Sums and products sin C + sin D = 2 sin C + D 2 cos C − D 2 \sin C + \sin D = 2\sin\dfrac{C + D}{2}\cos\dfrac{C - D}{2} sin C + sin D = 2 sin 2 C + D cos 2 C − D sin C − sin D = 2 cos C + D 2 sin C − D 2 \sin C - \sin D = 2\cos\dfrac{C + D}{2}\sin\dfrac{C - D}{2} sin C − sin D = 2 cos 2 C + D sin 2 C − D cos C + cos D = 2 cos C + D 2 cos C − D 2 \cos C + \cos D = 2\cos\dfrac{C + D}{2}\cos\dfrac{C - D}{2} cos C + cos D = 2 cos 2 C + D cos 2 C − D cos C − cos D = − 2 sin C + D 2 sin C − D 2 \cos C - \cos D = -2\sin\dfrac{C + D}{2}\sin\dfrac{C - D}{2} cos C − cos D = − 2 sin 2 C + D sin 2 C − D 2 sin x sin y = cos ( x − y ) − cos ( x + y ) 2\sin x \sin y = \cos(x - y) - \cos(x + y) 2 sin x sin y = cos ( x − y ) − cos ( x + y ) , and the other products the same way A worked example Simplify sin x + sin 3 x cos x + cos 3 x \dfrac{\sin x + \sin 3x}{\cos x + \cos 3x} cos x + cos 3 x sin x + sin 3 x .
The top. sin 3 x + sin x = 2 sin 2 x cos x \sin 3x + \sin x = 2\sin 2x \cos x sin 3 x + sin x = 2 sin 2 x cos x . The bottom. cos 3 x + cos x = 2 cos 2 x cos x \cos 3x + \cos x = 2\cos 2x \cos x cos 3 x + cos x = 2 cos 2 x cos x . Cancel 2 cos x 2\cos x 2 cos x : the result is tan 2 x \tan 2x tan 2 x . Practice with new numbers
A program to try A true identity holds at every angle. The program tests five rules at two hundred random angles and counts the misses.
Run it. Which two rules are false?
1. Add the rule tan 2 x = 2 tan x 1 − tan 2 x \tan 2x = \dfrac{2\tan x}{1 - \tan^2 x} tan 2 x = 1 − tan 2 x 2 tan x . Does it ever miss?
Mistakes to avoid Splitting over a sum. cos ( x + y ) ≠ cos x + cos y \cos(x + y) \neq \cos x + \cos y cos ( x + y ) = cos x + cos y , and sin 2 x ≠ 2 sin x \sin 2x \neq 2\sin x sin 2 x = 2 sin x . The cosine sign. cos ( x + y ) \cos(x + y) cos ( x + y ) has a minus sign; cos ( x − y ) \cos(x - y) cos ( x − y ) has a plus sign. The quadrant. sin 2 + cos 2 = 1 \sin^2 + \cos^2 = 1 sin 2 + cos 2 = 1 gives a size; the quadrant gives the sign. Half angles. The sign comes from the quadrant of x 2 \dfrac{x}{2} 2 x , not of x x x . The halves. In sin C + sin D \sin C + \sin D sin C + sin D , the angles are half the sum and half the difference. Check yourself Find the exact value of cos 105 ∘ \cos 105^\circ cos 10 5 ∘ . x x x is acute and sin x = 5 13 \sin x = \dfrac{5}{13} sin x = 13 5 . Find cos 2 x \cos 2x cos 2 x . Write sin 7 x − sin 3 x \sin 7x - \sin 3x sin 7 x − sin 3 x as a product. For 0 < x < π 2 0 < x < \dfrac{\pi}{2} 0 < x < 2 π , which is larger: sin 2 x \sin 2x sin 2 x or 2 sin x 2\sin x 2 sin x ? Show that cos 2 x − cos 2 x = sin 2 x \cos^2 x - \cos 2x = \sin^2 x cos 2 x − cos 2 x = sin 2 x . 
Written by StemPy Lab