Allied angles are mirror images.

A point P at an acute angle x on the unit circle and its image under π − x, π + x, 2π − x or π/2 ± x (mirror in y-axis, half-turn, mirror in x-axis, quarter-turn); the coordinates show which function keeps its name and which sign flips.

Choose the allied angle for which sin stays the same and cos changes sign (π − x), then the one that swaps sin and cos with cos turning negative (π/2 + x).

Swapping sin and cos in π ± x; giving the sign from the acute angle alone.

• 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.

Teaching this idea? Put the simulation on your own page or LMS with this code.