Trigonometry

Sine, Cosine and Tangent Explained

For an acute angle in a right triangle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. The unit circle extends these functions to every real angle.

Reviewed 2026-07-26

Right-triangle ratios

Choose one acute angle as the reference angle. The opposite and adjacent sides are named relative to that angle, while the hypotenuse always lies opposite the right angle.

All right triangles with the same acute angle are similar, so their side-length ratios are constant even when the triangle is scaled.

Formulasin(theta)=opposite/hypotenuse; cos(theta)=adjacent/hypotenuse; tan(theta)=opposite/adjacent

The unit-circle definition

On a unit circle, the point reached by an angle theta has coordinates (cos(theta), sin(theta)). This definition handles angles larger than 90 degrees, negative angles, and repeated turns.

Because the radius is 1, both sine and cosine stay between -1 and 1.

Tangent and undefined angles

Tangent equals sin(theta)/cos(theta). It is undefined whenever cosine is zero, including 90 degrees plus any integer multiple of 180 degrees.

Near those angles a floating-point calculator may display a very large finite approximation rather than a mathematical infinity. Check the angle before interpreting the output.

Formulatan(theta) = sin(theta)/cos(theta), when cos(theta) != 0

Where this idea is used

Finding inaccessible heights and distances.
Resolving vectors into horizontal and vertical components.
Modeling waves, rotations, and periodic motion.

Worked examples

Find the ratios in a 3-4-5 triangle

Let the side opposite theta be 3 and the adjacent side be 4.

hypotenuse = 5
sin(theta)=3/5
cos(theta)=4/5
tan(theta)=3/4

Result: sin=0.6, cos=0.8, tan=0.75

Use a known angle

For theta = 30 degrees, the unit-circle point is (sqrt(3)/2, 1/2).

cos(30 degrees)=sqrt(3)/2
sin(30 degrees)=1/2
tan=(1/2)/(sqrt(3)/2)

Result: tan(30 degrees)=1/sqrt(3)=sqrt(3)/3

Common mistakes

Naming opposite and adjacent sides without reference to the chosen angle.
Using degree inputs in RAD mode.
Assuming tangent is defined when cosine is zero.

Questions about sine, cosine, and tangent

Can sine or cosine be greater than 1?

Not for real angles because they are coordinates on a unit circle.

Why can tangent be greater than 1?

It is a ratio of two coordinates and is not bounded like either coordinate individually.

Do triangle definitions work for obtuse angles?

The unit-circle definition is used; a right-triangle reference angle helps recover magnitudes and signs.