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Trigonometry overview (sin ,cos, tan)

The purpose of this page is to give just enough knowledge of trigonometry to allow formulas that appear in geometry to be evaluated. Only sin, cos and tan functions and their inverses are discussed here.

For a full discussion see The six trigonometric functions.

Angle units - degrees and radians

All three functions (sin,cos, tan) take an angle and give another number based on that angle. The angle can be measured in degrees or radians. There are 360 degrees in a full circle, and approximately 6.284 radians in a full circle (actually two times Pi).

In the formulas given on geometry pages the angles are usually in degrees. For geometry problems in degrees, make sure your calculator is in degrees mode. This is the most common reason for strange answers.

SIN

This is the 'sine' function. It takes a number x, which represents an angle, and gives another number, which is called the 'sine of x', or just 'sine x'. The angle x can be in degrees or radians (see units discussion above). For more on this see The sine function.

Using a calculator:
  1. First determine what units are being used for x. If you are solving a formula given on other pages (example sides of a regular polygon) check whether it expects the angle to be in degrees or radians and set the calculator accordingly.
  2. HP calculators using RPN: Enter the angle and press sin
  3. Algebraic calculators: Press 'sin', then the angle, then '='.
Try this:
Calculate the sine of 1.5 radians.   You should get 0.997 approx.

COS

This is the 'cosine' function. It takes a number x, which represents an angle, and gives another number, which is called the 'cosine of x', or just 'cosine x'. The angle x can be in degrees or radians (see units discussion above). For more on this see The cosine function.

Using a calculator:
  1. First determine what units are being used for x. If you are solving a formula given on other pages (example sides of a regular polygon) check whether it expects the angle to be in degrees or radians and set the calculator accordingly.
  2. HP calculators using RPN: Enter the angle and press cos
  3. Algebraic calculators: Press 'cos', then the angle, then '='.
Try this:
Calculate the cosine of 45 degrees.   You should get 0.707 approx.

TAN

This is the 'tangent' function. It takes a number x, which represents an angle, and gives another number, which is called the 'tangent of x', or just 'tan x'. The angle x can be in degrees or radians (see units discussion above). For more on this see The tangent function.

Using a calculator:
  1. First determine what units are being used for x. If you are solving a formula given on other pages (example sides of a regular polygon) check whether it expects the angle to be in degrees or radians and set the calculator accordingly.
  2. HP calculators using RPN: Enter the angle and press tan
  3. Algebraic calculators: Press 'tan', then the angle, then '='.
Try this:
Calculate the tangent of 80°.   You should get 5.67 approx.

Inverse functions

The three functions above each have corresponding inverse functions. They have the same names but with 'arc' in front. Just as sin x gives the sine of x, so arcsin x gives you the angle whose sin is x; the function goes the other way.

For more on this see Inverse trigonometric functions.

So for example the sin of 50 degrees is 0.766.
The arcsin of 0.766 is 50 degrees.

The three inverse functions are

arcsin xgives the angle whose sin is x
arcscos xgives the angle whose cos is x
arctan xgives the angle whose tan is x

Other trigonometry topics

Angles

Trigonometric functions

Solving trigonometry problems

Calculus