Trigonometric Ratios of Allied Angles: Rules and Table

Allied angles are angles related to θ by adding or subtracting multiples of 90°, such as 90° − θ, 180° + θ or 360° − θ. Their trigonometric ratios can always be rewritten as a ratio of the acute angle θ, with a possible sign change, which lets you evaluate sines and cosines of any angle using only acute-angle values.

This matters because real problems hand you angles like 150°, 210° or 330°, far outside the first quadrant. Allied-angle rules reduce every one of them to a familiar acute angle, so a single table of values handles the whole circle.

Key Concepts

• Allied angles: 90° ± θ, 180° ± θ, 270° ± θ and 360° − θ (equivalently π/2 ± θ, π ± θ, 3π/2 ± θ, 2π − θ).
• Change or stay: for 90° ± θ and 270° ± θ the function changes (sin ↔ cos, tan ↔ cot, sec ↔ csc); for 180° ± θ and 360° − θ it stays the same.
• ASTC rule: the sign of the result follows the quadrant of the allied angle: All positive in Q1, Sine positive in Q2, Tangent positive in Q3, Cosine positive in Q4.
• Reference angle: the acute angle θ that the allied angle reduces to.

Formulas

• Complementary: sin(90° − θ) = cos θ; cos(90° − θ) = sin θ; tan(90° − θ) = cot θ.
• Supplementary: sin(180° − θ) = sin θ; cos(180° − θ) = −cos θ; tan(180° − θ) = −tan θ.
• Third quadrant: sin(180° + θ) = −sin θ; cos(180° + θ) = −cos θ; tan(180° + θ) = tan θ.
• Full circle: sin(360° − θ) = −sin θ; cos(360° − θ) = cos θ; tan(360° − θ) = −tan θ.

Worked Example

Evaluate sin 135° + cos 225° exactly.

1. Write 135° as an allied angle: 135° = 180° − 45°. The function stays sine, and 135° lies in Q2 where sine is positive, so sin 135° = sin 45° = √2/2.
2. Write 225° as 180° + 45°. The function stays cosine, and 225° lies in Q3 where cosine is negative, so cos 225° = −cos 45° = −√2/2.
3. Add them: √2/2 + (−√2/2) = 0.
4. Numerical check: √2/2 ≈ 0.7071, and 0.7071 − 0.7071 = 0, confirming the exact result.

Common Misconceptions

• 180° does not change the function. Only 90° and 270° (odd multiples of 90°) swap sine with cosine.
• The sign comes from the allied angle's quadrant, not from θ itself. Always ask "which quadrant is 180° + θ in?" before choosing the sign.
• These rules apply only to allied forms. An angle like 60° + θ is not allied and cannot be reduced this way.

Real-World Applications

• Simplifying expressions: reducing sin(180° + θ) or cos(270° − θ) to single ratios of θ.
• Solving trig equations: finding all angles with a given sine or cosine value.
• Physics: shifting wave phases by quarter or half cycles in oscillations and AC circuits.
• Navigation: converting bearings, which are measured from north, into standard position angles.

Allied angles reduce any angle to an acute reference angle: the function changes for 90° and 270° but stays for 180° and 360°, and the sign follows the ASTC quadrant rule. Apply change-or-stay first, quadrant sign second, and the whole circle collapses to the first quadrant.