Finding values

One ratio and the quadrant give all six. Use Pythagoras for the size, and the quadrant for the sign.

Example: sin x = –7/25 with x in the third quadrant.

• cos²x = 1 – 49/625 = 576/625, so cos x = ±24/25. In quadrant III cos is negative: cos x = –24/25.
• tan x = sin x/cos x = 7/24 (positive in quadrant III, as it should be), so cot x = 24/7.
• sec x = –25/24 and cosec x = –25/7.

Angles beyond 90°: the allied-angle method. Write the angle as 90°, 180°, 270° or 360° plus or minus an acute angle θ. Then:

• The sign comes from the quadrant the whole angle lies in.
• With 180° or 360°, the function stays the same.
• With 90° or 270°, it switches to its partner: sin ↔ cos, tan ↔ cot, sec ↔ cosec.
• Using 180° or 360° where you can avoids the switch.

Examples:

• sin 150° = sin(180° – 30°): quadrant II, sin positive, so +sin 30° = 1/2.
• cos 210° = cos(180° + 30°): quadrant III, cos negative, so –cos 30° = –√3/2.
• tan 240° = tan(180° + 60°): quadrant III, tan positive, so tan 60° = √3.
• sin 300° = sin(360° – 60°): quadrant IV, sin negative, so –sin 60° = –√3/2.

Big angles: sin and cos repeat every 360° (2π), so remove any multiple of 360°. tan and cot repeat every 180° (π), so for them any multiple of 180° can go.

• cos 750° = cos(750° – 720°) = cos 30° = √3/2.
• sin 17π/6 = sin(17π/6 – 2π) = sin 5π/6 = sin(π – π/6) = 1/2.
• tan 13π/4 = tan(13π/4 – 3π) = tan π/4 = 1.
• sin(–600°) = –sin 600° = –sin 240° = –(–√3/2) = √3/2: first take the minus out, then reduce.

In radians, a handy test is the number in front of π. If it is more than 2, subtract an even multiple of π first. If it is less than 1, write the angle as π minus something, as with 5π/6 above.

Two shortcuts save a lot of work:

• If A + B = 90°, then sin A = cos B and tan A = cot B. So sin 70° = cos 20°, and sin²20° + sin²70° = sin²20° + cos²20° = 1.
• If A + B = 180°, then sin A = sin B but cos A = –cos B. So cos 130° + cos 50° = 0, without knowing either value.