On this page
The triple-angle formulas are easy to write down and easy to mix up. Students often learn them as three lines to memorise and then do not recognise when a problem needs one. This post derives each formula in two lines, then shows the three jobs they do in exam problems: exact values, cubic equations and reducing powers.
Where are these in the syllabus?
The NCERT Class 11 chapter Trigonometric Functions (Reprint 2026-27) proves sin 3x, cos 3x and tan 3x as identities 17 to 19, and uses tan 3x in a worked example. The CBSE 2026-27 mathematics curriculum lists "Identities related to sin 2x, cos 2x, tan 2x, sin 3x, cos 3x and tan 3x". The JEE (Main) 2026 syllabus lists trigonometrical identities, and the JEE (Advanced) 2026 brochure lists "formulae involving multiple and sub-multiple angles". Check your exam year's document.
How do you derive them?
NCERT's method: write 3x = 2x + x and use the addition formula.
sin 3x = sin 2x cos x + cos 2x sin x = 2 sin x cos²x + (1 − 2 sin²x) sin x. Replace cos²x with 1 − sin²x: 2 sin x − 2 sin³x + sin x − 2 sin³x = 3 sin x − 4 sin³x.
cos 3x = cos 2x cos x − sin 2x sin x = (2 cos²x − 1) cos x − 2 sin²x cos x. Replace sin²x with 1 − cos²x: 4 cos³x − 3 cos x.
tan 3x = (tan 2x + tan x)/(1 − tan 2x tan x), which simplifies to (3 tan x − tan³x)/(1 − 3 tan²x). NCERT adds the condition 3x ≠ nπ + π/2.
Job 1: exact values, such as sin 18°
Let θ = 18°. Then 5θ = 90°, so 2θ = 90° − 3θ, and sin 2θ = cos 3θ.
2 sin θ cos θ = 4 cos³θ − 3 cos θ.
cos θ ≠ 0, so divide by it: 2 sin θ = 4 cos²θ − 3 = 4(1 − sin²θ) − 3 = 1 − 4 sin²θ.
So 4 sin²θ + 2 sin θ − 1 = 0, a quadratic in s = sin θ:
s = (−2 ± √(4 + 16))/8 = (−1 ± √5)/4.
sin 18° is positive, so sin 18° = (√5 − 1)/4 ≈ 0.3090. A calculator gives 0.30902, which matches.
Job 2: cubic equations, such as one with root cos 20°
Put x = 20° in cos 3x = 4 cos³x − 3 cos x. Since cos 60° = 1/2:
4c³ − 3c = 1/2, or 8c³ − 6c − 1 = 0, where c = cos 20°.
The same equation holds for any angle whose triple has cosine 1/2: 3x = 60°, 420° or 780°, giving x = 20°, 140° and 260°. So the three roots are cos 20°, cos 140° and cos 260°. The cubic has no c² term, so the roots add to 0:
cos 20° + cos 140° + cos 260° = 0.
Since cos 140° = −cos 40° and cos 260° = −cos 80°, this gives a neat result: cos 20° = cos 40° + cos 80°. Numerically, 0.9397 = 0.7660 + 0.1736.
Job 3: reducing cubes before integrating
Rearrange sin 3x = 3 sin x − 4 sin³x:
sin³x = (3 sin x − sin 3x)/4, and similarly cos³x = (3 cos x + cos 3x)/4.
NCERT's Integrals chapter uses exactly this to integrate sin³x. Try a definite version: the integral of sin³x from 0 to π/2.
= (3/4) × integral of sin x − (1/4) × integral of sin 3x, both from 0 to π/2.
Integral of sin x from 0 to π/2 = 1. Integral of sin 3x from 0 to π/2 = [−cos 3x/3] = (−cos(3π/2) + cos 0)/3 = 1/3.
So the answer is 3/4 − 1/12 = 2/3. Our post on integrating powers of sin and cos has the other methods.
A quick tan 3x example
If tan x = 1/2, then tan 3x = (3/2 − 1/8)/(1 − 3/4) = (11/8)/(1/4) = 11/2. Check that 1 − 3 tan²x is not zero first; when tan²x = 1/3, tan 3x is undefined, which is NCERT's condition in another form.
Which mistakes cost marks?
Writing sin 3x = 3 sin x. Test with x = 30°: sin 90° = 1, but 3 sin 30° = 1.5.
Swapping the sine and cosine patterns. Use the x = 90° check in the callout, or x = 0° for cosine: cos 0° = 1 and 4 − 3 = 1.
Dividing by cos θ without saying it is non-zero. In the sin 18° proof, say why you can.
Keeping both roots. (−1 − √5)/4 is negative, so it cannot be sin 18°.
Forgetting the tan 3x condition. State 1 − 3 tan²x ≠ 0.
For the identity families that sit around these, see our trigonometric identities practice plan.
A practice routine
Derive all three formulas from 3x = 2x + x on a blank page, three times this week.
Use the 5θ = 90° idea to find cos 36°, starting from sin 2θ = cos 3θ with θ = 18° and then cos 2θ.
Write the cubic with root cos 40° and check one root on a calculator.
Integrate cos³x from 0 to π/2 using the reduction above.
When a trig proof stalls and you cannot see which identity to use, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer, so you can see why each identity is chosen.
Today, find cos 36° exactly and check that it equals (√5 + 1)/4.
