# Explaining Trigonometric Ratios: cos

By Kathleen Knowles, 03 Apr 2021

Trigonometry examines the relationship between the sides of a triangle, more specifically, right triangles. A right triangle has a 90° angle. The equations and ratios that describe the relationship between the sides of a triangle and its angles are trigonometric functions. In this particular article, we're going to explain one specific ratio: "cos" or cosine. But before we dive into cosine, let's take a look at the other ratios in trigonometry.

## Fundamental Trigonometric Functions

When we define the trigonometric ratios, let us define a right-angled triangle with one of the angles named *x. *This angle is 90°. You define the sides of a triangle as *a, b, *and *c* where *a* is the side adjacent to *x* and *b* is the side opposite *x*. *c* is the hypotenuse or the side opposite the right angle. There are six fundamental trigonometric functions.

*Sin x*is the ratio of the opposite side to the hypotenuse.`sin x = (opposite) / (hypotenuse) = b / c`

*Cos x*is the ratio of the adjacent side to the hypotenuse.`cos x = (adjacent) / (hypotenuse) = a / c`

*Tan x*is the opposite side to the adjacent side.`tan x = (opposite) / (adjacent) = b / a`

- If you do
`(b / c) / (a / c)`

*,*you will get`b/a`

which is*tan x.*So*tan x*can be expressed as the ratio of sin to cos.`tan x = sin x / cos x`

.

*Cosec x*is the reciprocal of*sin x*`csc x = 1 / sin x`

*Sec x*, is the reciprocal of*cos x.*`sec x = 1 / cos x`

*Cot x*is the reciprocal of*tan x*`cot x = 1 / tan x`

Out of the six fundamental trigonometric functions, you will mostly be concerned with sin, cos, and tan.

## Cosine Function

You can define a cosine function using a right-angled triangle as defined above. However, you can use cosine in several other applications.

### Defining Cosine using Differential Equations

You can use the cosine using differential equations. The cos and sin are the two differentiable trig functions and they have a special relationship.

* cos x = ( d / dx ) sin x * and

`-sin x = ( d / dx ) cos x`

The above definitions are useful when solving differential equations. Both of the above expressions are solutions to the differential equation:

`y” + y = 0`

### The Power Series Expansion

Trigonometric functions are also defined using power series. By applying the Taylor series to cosine, you can obtain another definition.

`cos x = 1 – ( x2 / 2! ) + ( x4 / 4! ) – ( x6 / 6! )`

…..

### Exponential Expression using Euler's Formula

Euler had related the sine and cosine functions by the expression:

`ejx = cos x + j sin x`

`e-jx = cos x – j sinx`

The *j *in the above expressions refers to the imaginary unit, which is equivalent to the square root of (-1). Euler's expression or relationship is true for all complex values. This means that the formula is true for all real values of *x*.

If we add the above equations, we can find a concise expression for cos x in the complex domain as:

`cos x = ( ejx + e-jx ) / 2`

If the value of *x* is real, you can write the expression as:

`cos x = Re( ejx )`

### Values of Cosine in the Four Quadrants of a Circle

Since a full circle is 360°, you can express the cosine in different parts of a circle starting at 0° up to 360°. In the first quadrant of a circle, angles from 0° to 90°, the value of cos is positive. In the second quadrant with a range of angles from 90° to 180°, the value of cos is negative. In the third quadrant with a range of angles from 180° to 270°, the value of cos is still negative. In the fourth quadrant, with the range of angles from 270° to 360°, the value of cos is positive.

## Examples of Using Cosines

Before I proceed, let me introduce a trigonometric identity. Trigonometric identities are relationships between the trigonometric functions which are true at all conditions. One of them is `cos2 x + sin2 x = 1`

*. *Let's look at a few examples and apply this trigonometric identity.

### Example 1

A right triangle has a sin of 0.866. Find the cosine of the angle.

Taking our trigonometric identity, we can rearrange the expression.

`cos2 x = 1 – sin2 x`

`cos x = ( 1 – sin2 x )1/2`

Since we know the value of *sin x*, let us substitute it for *sin2 x* in the expression.

`cos x = ( 1 – sin2 x )1/2`

`cos x = (1 – 0.8662 )1/2`

`cos x = 0.5`

### Example 2

A right triangle (ABC) has a right angle at B. The length of the hypotenuse, AC, is 5cm and the side BC is 3 cm. Find the angle at C.

To refresh your memory, the cosine of an angle is *adjacent/hypotenuse*. Let the angle at C be *x*.

`cos x = 3 / 5`

`x = cos-1 ( 3 / 5 )`

*x = 53*°

The angle at C is 53°.

The expression *cos-1* means the cos inverse. It is the inverse of the cos function. If the cosine of an angle is *x*, then cos-1 *x* is the original angle.

`cos 60° = 0.5`

`cos-1 0.5 = 60°`

### Example 3

Find the cosine of the following angles using our circle quadrants.

- 660°
- 234°
- -60°

660° is bigger than a circle, which is 360°. But since an angle is a degree of turning, it means that the point has moved a full circle and then some. The full circle will not count since the angle of interest is the amount it turned from the starting point to the final point.

So `cos 660° = cos ( 660 – 360 )° = cos 300°`

Since 300° falls within the fourth quadrant, it means that the value of cos is positive.

`cos 300° = 0.5`

For the second question, 234° is less than 360° so the moving point has not moved a full circle. Also, 234° falls within the third quadrant. Therefore, the value of cos is negative.

`cos 234° = -0.588`

The third problem has a negative angle of -60°. Negative angles mean that the direction of movement is clockwise instead of the normal anticlockwise. So if you move clockwise 60° you will end up in the fourth quadrant.

`-60° = ( 360 – 60 )° = 300°`

`cos 300° = 0.5`

## Final Thoughts

You can easily find the cos of an angle by looking it up in a cos table or by pressing *cos *and the angle on a scientific calculator. On most scientific calculators, the cos-1 inverse function is a second function of the cos, usually on the same key. For such calculators, to use the cos inverse function, press SHIFT and COS on the calculator.

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