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Cosine (cos) function - Trigonometry

In a right triangle, the cosine of an angle is the length of the adjacent side divided by the length of the hypotenuse.
Try this Drag any vertex of the triangle and see how the cosine of A and C are calculated.

The cosine function, along with sine and tangent, is one of the three most common trigonometric functions. In any right triangle, the cosine of an angle is the length of the adjacent side (A) divided by the length of the hypotenuse (H). In a formula, it is written simply as 'cos'.

As an example, let's say we want to find the cosine of angle C in the figure above (click 'reset' first). From the formula above we know that the cosine of an angle is the adjacent side divided by the hypotenuse. The adjacent side is BC and has a length of 26. The hypotenuse is AC with a length of 30. So we can write This division on the calculator comes out to 0.866. So we can say "The cosine of 30° is 0.866 " or

Use your calculator to find the cosine of 30°. It should come out to 0.8660 as above.
(If it doesn't - make sure the calculator is set to work in degrees and not radians).

Example - using cosine to find the hypotenuse

If we look at the general definition - we see that there are three variables: the measure of the angle x, and the lengths of the two sides (Adjacent and Hypotenuse). So if we have any two of them, we can find the third.

In the figure above, click 'reset'. Imagine we didn't know the length of the hypotenuse H. We know that the cosine of A (60°) is the adjacent side (15) divided by H. From our calculator we find that cos60 is 0.5, so we can write Transposing: which comes out to 30, which matches the figure above.

The inverse cosine function - arccos

For every trigonometry function such as cos, there is an inverse function that works in reverse. These inverse functions have the same name but with 'arc' in front. So the inverse of cos is arccos etc.

When we see "arccos A", we interpret it as "the angle whose cosine is A"

cos60 = 0.5 Means: The cosine of 60 degrees is 0.5
arccos0.5 = 60 Means: The angle whose cosine is 0.5 is 60 degrees.

We use it when we know what the cosine of an angle is, and want to know the actual angle.

See also Arccosine definition and Inverse functions - trigonometry

Large and negative angles

In a right triangle, the two variable angles are always less than 90° (See Interior angles of a triangle). But we can in fact find the cosine of any angle, no matter how large, and also the cosine of negative angles. For more on this see Functions of large and negative angles.

Graphing the cosine function

When the cosine of an angle is graphed against the angle, the result is a shape similar to that on the right.

For more on this see Graphing the cosine function.

The derivative of cos(x)

In calculus, the derivative of cos(x) is –sin(x). This means that at any value of x, the rate of change or slope of cos(x) is –sin(x). For more on this see Derivatives of trigonometric functions together with the derivatives of other trig functions. See also the Calculus Table of Contents.

Pages referring to 'cosine'

This page explores the derivatives of trigonometric functions in calculus. Interactive calculus applet.
Interactive demonstration of the graph of the cosine function in trigonometry
Introduction to the 6 trigonometry functions - sine, cosine, tangent, secant, cosecant, cotangent
A step by step proof of the Law of Cosines
A memory aid for remembering the definitions of sin, cos, and tan
An introduction the the concept of solving a triangle.
How to use the Law of Cosines to solve a triangle given two sides and included angle
Definition of the arccos function in trigonometry. The inverse of the cosine function. The angle whose cosine is a given number.