Sin Cos Formulas — Full List, Identities, Examples

TL;DR

The sin cos formulas are the family of trigonometric identities relating sine and cosine — the Pythagorean identity, the sum and difference formulas, the double-angle and half-angle identities, and the product-to-sum conversions. This article organises all of them into one map, proves the ones the others are built from, works six examples in degrees and radians, and shows which two formulas you actually have to memorise.

Last updated on June 22, 2026 8 min read

What Are the Sin Cos Formulas?

The sin cos formulas are the set of trigonometric identities that express relationships between the sine and cosine functions — and, through them, every other trig function. At the foundation is the right-triangle definition: for an acute angle θ,

[ \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\sin\theta}{\cos\theta}. ]

On the unit circle, a point at angle θ has coordinates exactly (cos⁡θ,sin⁡θ) — cosine is the x-coordinate, sine is the y-coordinate. These two anchors, the triangle and the circle, are the same fact in two costumes, and every formula below respects both.

The Pythagorean Identity — The One You Must Know Cold

Everything starts here:

[ \sin^2\theta + \cos^2\theta = 1 ]

It is the Pythagorean theorem written in trig: on the unit circle the point (cos⁡θ,sin⁡θ) sits at distance 1 from the origin, so x²+y²=1 becomes ( \cos^2\theta + \sin^2\theta = 1 ). Dividing through by ( \cos^2\theta ) gives ( 1 + \tan^2\theta = \sec^2\theta ); dividing by ( \sin^2\theta ) gives ( 1 + \cot^2\theta = \csc^2\theta ). Our Pythagorean identities page proves all three in full. If you know this one identity cold, you can rebuild the cosine-from-sine relationship any time: ( \cos\theta = \pm\sqrt{1 - \sin^2\theta} ).

Sum and Difference Formulas

These give the sine and cosine of a combined angle:

[ \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B ]

[ \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B ]

Read the cosine sign carefully: the sign between the two products is the opposite of the sign between the angles (( \cos(A+B) ) takes a minus). Sine keeps the same sign. These are the formulas the double- and half-angle identities are built from.

Double Angle Formulas

Set A=B=θ in the sum formulas and the double-angle identities drop out:

[ \sin 2\theta = 2 \sin \theta \cos \theta ]

[ \cos 2\theta = \cos^2\theta - \sin^2\theta = 2 \cos^2\theta - 1 = 1 - 2 \sin^2\theta ]

Cosine has three equivalent forms because ( \sin^2\theta + \cos^2\theta = 1 ) lets you trade one squared term for the other. The companion sin 2x derivation and cos2x formula pages each work one of these in depth.

Half Angle Formulas

Solve the cosine double-angle forms backwards ( (\theta \to \frac{\theta}{2}) ) and you get the half-angle identities:

[ \sin \frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}, \quad \cos \frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}} ]

The ± is fixed by the quadrant of ( \frac{\theta}{2} ) — see the full half angle formula walk-through, including the sign chart.

Product-to-Sum and Sum-to-Product Formulas

Adding and subtracting the sum and difference formulas turns products into sums:

[ \sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)] ]

[ \cos A \cos B = \frac{1}{2}[\cos(A-B) + \cos(A+B)] ]

Running the algebra the other direction gives the sum-to-product forms, e.g. ( \sin A + \sin B = 2 \sin \frac{A+B}{2} \cos \frac{A-B}{2} ).

A Quick-Reference Table

The full identity catalogue, including reciprocal and cofunction relations, lives in our trigonometric identities reference.

Examples of Sin Cos Formulas

Example 1

Given ( \sin\theta = \frac{3}{5} ) with θ in Quadrant I, find ( \cos\theta ).

Use the Pythagorean identity, solved for cosine. In Quadrant I cosine is positive:

[ \cos\theta = \sqrt{1 - \sin^2\theta} = \sqrt{1 - \frac{9}{25}} = \frac{4}{5}. ]

Final answer: ( \cos\theta = \frac{4}{5} ).

Example 2

Given ( \sin\theta = \frac{3}{5} ) with θ in Quadrant II, find ( \cos\theta ).

A natural reflex is to repeat Example 1 exactly and report ( \cos\theta = \frac{4}{5} ).

Wrong path. Writing ( \cos\theta = +\sqrt{1 - \frac{9}{25}} = +\frac{4}{5} ). The magnitude is right, but the sign is wrong: in Quadrant II, sine is positive while cosine is negative. Quadrant II forces the negative root:

[ \cos\theta = -\sqrt{1 - \frac{9}{25}} = -\frac{4}{5}. ]

Final answer: ( \cos\theta = -\frac{4}{5} ).

Example 3

Compute ( \sin 75° ) exactly using a sum formula.

Write 75°=45°+30° and apply ( \sin(A+B) ):

[ \sin 75° = \sin 45°\cos 30° + \cos 45°\sin 30° = \frac{\sqrt{6} + \sqrt{2}}{4}. ]

Final answer: ( \sin 75° = \frac{\sqrt{6} + \sqrt{2}}{4} \approx 0.966. )**

Example 4

Given ( \sin\theta = \frac{12}{13} ) with ( \theta ) acute, find ( \sin 2\theta ) and ( \cos 2\theta ).

First ( \cos\theta = \sqrt{1 - \frac{144}{169}} = \frac{5}{13} ). Then apply ( \sin 2\theta = 2\sin\theta\cos\theta ):

[ \sin 2\theta = 2\cdot\frac{12}{13}\cdot\frac{5}{13} = \frac{120}{169}. ]

[ \cos 2\theta = 1 - 2\sin^2\theta = 1 - 2\cdot\frac{144}{169} = -\frac{119}{169}. ]

Final answer: ( \sin 2\theta = \frac{120}{169}, \quad \cos 2\theta = -\frac{119}{169}. )

Example 5

Verify ( \cos 2\theta = 2\cos^2\theta - 1 ) at ( \theta = 60°).

Left side: ( \cos 120° = -\frac{1}{2} ). Right side: ( 2\cos^2 60° - 1 = -\frac{1}{2} ). Final answer: identity verified, ( -\frac{1}{2} = -\frac{1}{2}. )

Example 6

Rewrite ( 2\sin 3\theta \cos\theta ) as a sum.

Use the product-to-sum formula:

[ 2\sin 3\theta\cos\theta = \sin(4\theta) + \sin(2\theta). ]

Final answer: ( 2\sin 3\theta\cos\theta = \sin 4\theta + \sin 2\theta. )

Where Sin and Cos Formulas Carry Real Weight

Sine and cosine are how every periodic thing in the universe gets described, and these identities are the algebra that makes the description usable.

• Sound and music — beats. When two notes of nearly equal frequency play together, the sum-to-product formula rewrites ( \sin(\omega_1 t) + \sin(\omega_2 t) ) as a single tone amplitude-modulated at the difference frequency.

• Electrical engineering — AC power. Alternating current is ( V_0\cos(\omega t) ); computing power means multiplying two cosines, and the product-to-sum identity is what separates the average (real) power from the oscillating part.

• Optics and interferometry. Overlapping light waves combine through the sum formulas — the foundation of everything from anti-reflective coatings to LIGO's gravitational-wave detection.

• Computer graphics. Rotating a point by angle θ multiplies it by a matrix of ( \sin \theta ) and ( \cos \theta );

For Class 11 students, the immediate payoff is exact-value computation and identity proofs — but the same five families reappear in calculus, physics, and signal processing.

Tripping Points to Avoid

Mistake 1: Writing (sin θ)² as sin θ²

Where it slips in: Reading or writing ( \sin^2\theta ) as ( \sin(\theta^2) ).

Don't do this: Square the angle. ( \sin^2\theta ) squares the output of sine, not the input.

Mistake 2: Confusing sin²θ + cos²θ = 1 with (sin θ + cos θ)² = 1

Where it slips in: Squaring the sum of sine and cosine and expecting 1.

Mistake 3: Copying the cosine sum sign from the angle operation

Where it slips in: Writing ( \cos(A+B) = \cos A \cos B + \sin A \sin B ) — matching the inner + to the expansion.

Conclusion

• The sin cos formulas are five families: Pythagorean, sum and difference, double angle, half angle, and product-to-sum.

• The Pythagorean identity ( \sin^2\theta + \cos^2\theta = 1 ) is the one to know cold — the cosine-from-sine relationship rebuilds from it.

• The sum and difference formulas generate the double- and half-angle identities by substitution.

• The most common errors are notation (( sin^2\theta ) vs ( sin(\theta^2) )), the missing cross term, and copying the cosine sum sign.