Lesson 9 of 13 · 8 min
Allied angles
NCERT §3.4
The wheel stops with Meera's cabin at 30°. Her friend Kabir's cabin is at 150°, and two more are at 210° and −30°. Their positions are mirror images of hers, so their sines and cosines must be close relatives of sin 30° and cos 30°.
The lesson in notes
In short
Choosing special values in the sum and difference formulas relates an angle to x: cos (π/2 + x) = −sin x and sin (π/2 + x) = cos x.
Across the y-axis: cos (π − x) = −cos x and sin (π − x) = sin x.
Through the origin: cos (π + x) = −cos x and sin (π + x) = −sin x, so tan (π + x) = tan x.
Across the x-axis again: cos (2π − x) = cos x and sin (2π − x) = −sin x.
A working rule: with π or 2π the function keeps its name; with π/2 (or 3π/2) sine and cosine swap names. The sign in front is the sign the original function has in the quadrant where the new angle lies, taking x acute.
Worked values: sin 120° = sin (180° − 60°) = √3/2; cos 210° = cos (180° + 30°) = −√3/2; tan 315° = tan (360° − 45°) = −1; sin 17π/6 = sin (2π + 5π/6) = sin 5π/6 = 1/2.
Similar results for tan, cot, sec and cosec follow from those for sin and cos, for example tan (π − x) = −tan x and sec (π + x) = −sec x.