Trigonometric Identities

Pick an identity, move the angles and watch a unit-circle construction prove it. Both sides are evaluated and plotted, so you can see they agree for every angle.

Intermediate

sin(α + β) = sin α cos β + cos α sin β

Geometric construction

Drag on the circle to change α.

α35°β25°

Both sides, evaluated

sin(α + β): 0.8660
sin α cos β: 0.5198
cos α sin β: 0.3462
sin α cos β + cos α sin β: 0.8660
difference: 0.0e+0

Both sides over a full turn

Wide blue band: left side. Dashed black line: right side. Dotted lines: the individual terms. β is held at 25°.

Walk from O along the α ray for a distance cos β to reach A, then turn 90° and walk sin β to reach B. B is on the unit circle at angle α + β. The height of A is sin α cos β (green) and the second leg adds cos α sin β (orange), so the height of B, sin(α + β), is their sum.

What is an identity?

An identity is an equation that holds for every value of the variable, not just for a few solutions. sin²θ + cos²θ = 1 is true at 0°, at 37°, at 250°, at any angle you choose. That is why the two curves in the graph above lie exactly on top of each other.

Every identity on this page comes from one picture: the unit circle, where the point at angle θ is (cos θ, sin θ). The Pythagorean identity is Pythagoras on the triangle under that point. The sum formulas come from reaching the point at angle α + β in two right-angled steps, and the double- and half-angle formulas are special cases.

Key formulas

Sum and difference

sin(α ± β) = sin α cos β ± cos α sin β
cos(α ± β) = cos α cos β ∓ sin α sin β

α, β: any two angles
±, ∓: take the top signs together or the bottom signs together

Family	Identity	Related forms
Pythagorean	sin²θ + cos²θ = 1	1 + tan²θ = sec²θ
Sum (sine)	sin(α + β) = sin α cos β + cos α sin β	sin(α − β) = sin α cos β − cos α sin β
Sum (cosine)	cos(α + β) = cos α cos β − sin α sin β	cos(α − β) = cos α cos β + sin α sin β
Double angle	sin 2α = 2 sin α cos α	cos 2α = cos²α − sin²α = 2cos²α − 1 = 1 − 2sin²α
Half angle	cos²(θ/2) = (1 + cos θ)/2	sin²(θ/2) = (1 − cos θ)/2, tan(θ/2) = sin θ/(1 + cos θ)

How to use this visualizer

1
Choose an identity from the row of buttons. The identity is shown in full above the figure.
2
Move the α (or θ) slider, or drag anywhere on the circle, to change the angle. For sum and difference identities, use the β slider too.
3
In the construction, the green and orange pieces are the two terms of the right-hand side. They stack along an axis to give the coordinate of B (or the length QF in the half-angle case).
4
Check the numeric panel: the left side and the sum of the terms agree to every displayed decimal place.
5
The graph sweeps the angle through a full turn. The blue band (left side) and the dashed line (right side) coincide everywhere, which is what makes it an identity.

Where identities are used

Simplifying expressions

Replace sin²x + cos²x with 1, or 2 sin x cos x with sin 2x, to shorten calculations.

Integration

Power-reduction forms such as cos²x = (1 + cos 2x)/2 turn hard integrals into easy ones.

Exact values

sin 75° = sin(45° + 30°) gives an exact surd instead of a decimal approximation.

Signals and rotation

The sum formulas are exactly the rotation matrix: rotating by α then β equals rotating by α + β.

Practice problems

Problem 1: An exact value

Use a sum formula to find the exact value of sin 75°.

Show solution

sin 75° = sin(45° + 30°)

= sin 45° cos 30° + cos 45° sin 30°

= (√2/2)(√3/2) + (√2/2)(1/2)

= (√6 + √2)/4

sin 75° = (√6 + √2)/4 ≈ 0.9659

Problem 2: Double angle from one ratio

Angle θ is in the first quadrant and sin θ = 3/5. Find sin 2θ and cos 2θ.

Show solution

cos²θ = 1 − sin²θ = 1 − 9/25 = 16/25, and θ is in quadrant I so cos θ = 4/5.

sin 2θ = 2 sin θ cos θ = 2 · (3/5) · (4/5) = 24/25

cos 2θ = cos²θ − sin²θ = 16/25 − 9/25 = 7/25

Check: (24/25)² + (7/25)² = (576 + 49)/625 = 1

sin 2θ = 24/25, cos 2θ = 7/25

Discussion

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