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Questions such as "find tan 300°" or "simplify sin(π − x) cos(π/2 + x)" appear in every Class 11 trigonometry test, and they are easy marks once two rules are automatic. The trouble is that students learn a long table by heart and then misremember one line. This post replaces the table with two rules, shows where they come from, and works through examples.
What does the syllabus include?
The CBSE Mathematics curriculum for Class XI (2026-27) includes the unit-circle definition of the trigonometric functions, their signs, and the formulas for sin(x ± y) and cos(x ± y) "and their simple applications". The JEE (Main) 2026 syllabus lists trigonometrical identities and functions.
In the NCERT Class 11 chapter on trigonometric functions, the allied-angle results come straight out of the sum and difference formulas. For example, cos(π − x) = −cos x, sin(π − x) = sin x, cos(π + x) = −cos x, sin(π + x) = −sin x, cos(π/2 + x) = −sin x and sin(π/2 + x) = cos x.
What are the two rules?
Write the angle as nπ/2 ± x (or n × 90° ± x), and for working out the sign, imagine x is acute.
Which ratio? If n is even (π, 2π, 180°, 360°), the ratio stays the same. If n is odd (π/2, 3π/2, 90°, 270°), it changes to its co-function: sin ↔ cos, tan ↔ cot, sec ↔ cosec.
Which sign? Find the quadrant where nπ/2 ± x lies. The sign is the sign of the original ratio in that quadrant. ASTC is a reminder: in quadrants I, II, III and IV, All, Sine, Tangent and Cosine (with their reciprocals) are positive.
Before either rule, deal with negative angles: sin(−x) = −sin x and cos(−x) = cos x. And remove whole turns, since sin and cos repeat every 2π (360°).
Worked examples in degrees
tan 300°. 300° = 360° − 60°, which is in quadrant IV, where tan is negative. 360° means n even, so tan stays tan. tan 300° = −tan 60° = −√3.
sin 1230°. Remove whole turns: 1230° − 3 × 360° = 150°. 150° = 180° − 30°, quadrant II, where sin is positive. sin 1230° = sin 30° = 1/2.
cos(−930°). cos is even, so this is cos 930°. 930° − 2 × 360° = 210° = 180° + 30°, quadrant III, where cos is negative. cos(−930°) = −cos 30° = −√3/2.
NCERT's own examples work the same way: sin(31π/3) = sin(10π + π/3) = √3/2, and cos(−1710°) = cos(90°) = 0 after adding 5 × 360°.
Worked example with radians and a check
Show that cos(3π/2 + x) = sin x.
By the rules: 3π/2 means n odd, so cos changes to sin. For acute x, 3π/2 + x lies in quadrant IV, where cos (the original ratio) is positive. So cos(3π/2 + x) = +sin x.
Check with the formula: cos(3π/2 + x) = cos(3π/2) cos x − sin(3π/2) sin x = 0 − (−1) sin x = sin x.
Now simplify E = sin(π − x) cos(π/2 + x) / [cos(π + x) sin(3π/2 − x)].
sin(π − x) = sin x
cos(π/2 + x) = −sin x
cos(π + x) = −cos x
sin(3π/2 − x) = −cos x (co-function, quadrant III, sin negative)
So E = (sin x)(−sin x) / [(−cos x)(−cos x)] = −sin²x/cos²x = −tan²x.
Which mistakes cost marks?
Taking the sign from the new ratio. For cos(π/2 + x), the angle is in quadrant II, where cos is negative, so the answer is −sin x, even though sin is positive there.
Changing the ratio for π or 2π. Only odd multiples of π/2 swap.
Forgetting sin(−x) = −sin x before reducing.
Mixing degrees and radians in one line.
Why do the rules beat a memorised table? Weber (2005) studied two college trigonometry courses and found that students taught in a lecture-based course developed a very limited understanding of trigonometric functions, while students given an experimental course, which included an activity on computing sines and cosines with the unit circle, developed a deep understanding. The two rules above are a short form of that unit-circle reasoning. Our post on trigonometric identities and equations builds on them.
If quadrants and signs still feel uncertain, Fermi's Concept Library covers every concept in physics, chemistry, biology, mathematics and computing.
A practice routine
For a week, do five reductions a day in mixed degrees and radians, saying "same or co-function, then sign" aloud for each.
Check one answer each day with the sum or difference formula.
Shuffle allied-angle questions in with other trigonometry. In Rohrer and Taylor (2007), students whose practice problems were randomly mixed did far better on a test a week later than students who practised one type at a time.
Finish with NCERT Exercise 3.2 questions 6 to 10.
Start with sin 765° and cosec(−1410°) from that exercise, using only the two rules.
