ScientificCalcul

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

Name sides relative to the chosen angle

Opposite and adjacent change when the reference angle changes. The hypotenuse is always opposite the right angle.

Right triangle side names A right triangle labels the horizontal side adjacent, the vertical side opposite, and the diagonal hypotenuse relative to angle theta. theta adjacentopposite hypotenuse
For the marked angle theta: sin(theta) = opposite/hypotenuse, cos(theta) = adjacent/hypotenuse, and tan(theta) = opposite/adjacent.

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

One angle, three related ratios

On the unit circle, cosine is the x-coordinate, sine is the y-coordinate, and tangent is their quotient when cosine is nonzero.

FunctionRight triangleUnit circleUndefined when
sin(theta)opposite / hypotenuseyNever for a real angle
cos(theta)adjacent / hypotenusexNever for a real angle
tan(theta)opposite / adjacenty/xcos(theta) = 0

Worked example 1

Find the ratios in a 3-4-5 triangle

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

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

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

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

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.

Sources and further reading