Trigonometric Functions and Allied Angles
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Direct answer
Every trigonometric ratio carries two pieces of information — a magnitude and a sign — and the sign is fixed by the quadrant, recalled as ASTC: All positive in the first, Sin in the second, Tan in the third, Cos in the fourth ("All Silver Tea Cups"). Allied-angle formulas handle rotations of the argument: sin(90° − θ) = cos θ, sin(180° − θ) = sin θ, sin(180° + θ) = −sin θ, with the working rule that odd multiples of 90° swap the function while even multiples keep it, the sign decided by ASTC. Compound-angle identities such as sin(A ± B) = sin A cos B ± cos A sin B extend the machinery to sums of angles.
What you must remember
- ASTC placement: All, Sin, Tan, Cos by quadrant — the mnemonic that fixes every sign error before it happens.
- Allied sets: sin(π − θ) = sin θ but cos(π − θ) = −cos θ; sin(π + θ) = −sin θ; sin(2π − θ) = −sin θ; sin(90° + θ) = cos θ, cos(90° − θ) = sin θ.
- Odd/even multiple rule: an odd multiple of 90° converts sine to cosine and vice versa; multiples of 180° preserve the function; the sign then follows the quadrant.
- Compound angles: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B — only the cosine sign flips.
- Exact values worth memory: sin 15° = (√6 − √2)/4; sin 18° = (√5 − 1)/4; cos 36° = (√5 + 1)/4 — derivable in seconds, but exam-ready recall saves minutes.
- Degree-radian discipline: 180° = π radians; a mixed-unit computation is an automatic error.
- Pythagorean triple: sin^2 θ + cos^2 θ = 1, 1 + tan^2 θ = sec^2 θ, 1 + cot^2 θ = cosec^2 θ.
Deriving sin 18 degrees
Five times 18° is 90°, so 2 × 18° and 3 × 18° are complementary: sin 36° = cos 54°. Expand both sides with double and triple angle formulas in terms of x = sin 18°: the left side is 2x cos 18°, the right side is 4cos^3 18° − 3cos 18°. Divide throughout by cos 18° (never zero): 2x = 4(1 − x^2) − 3, which tidies to 4x^2 + 2x − 1 = 0. The positive root is x = (√5 − 1)/4 ≈ 0.309. One bonus falls out free: cos 36° = 1 − 2sin^2 18° = (1 + √5)/4, the golden ratio over two — a number that reappears in pentagon geometry. The complement trick is the reusable part: whenever multiples of an unknown angle add to 90°, equate a function of one to the cofunction of the other.
Where marks leak
JEE Main tests quick evaluation — sin 150°, cos 225°, tan(−210°) — where the loss is always a sign, since sin 150° = sin 30° = 1/2 while cos 150° = −√3/2. JEE Advanced embeds allied angles inside larger structures: arguments of products of complex numbers, calculus substitutions, and multiple-angle equations. Three errors dominate: writing cos(A + B) = cos A + cos B (the plus-sign catastrophe), switching the function for an even multiple of 90° or keeping it for an odd multiple, and carrying degrees into a formula that assumed radians. The 18°-36°-54°-72° family deserves special respect: JEE has asked for sin 18° or cos 36° values often enough that the derivation above should be automatic.
Frequently asked questions
What is sin 150 degrees?
1/2, because sin(180° − 30°) = sin 30° and the second quadrant keeps sine positive.
What conversions happen at 90 + θ?
Sine becomes cosine: sin(90° + θ) = cos θ, while cos(90° + θ) = −sin θ.
What is the exact value of sin 18 degrees?
(√5 − 1)/4, roughly 0.309, derived from sin 36° = cos 54°.
What is cos(A − B)?
cos A cos B + sin A sin B; the minus sign belongs to cos(A + B).
Why memorise ASTC?
It fixes the sign of any ratio in any quadrant, which is where nearly all marks in allied-angle questions are lost.