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Explaining Trigonometric Ratios: cos

By Kathleen Cantor, 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 xc 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 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 = 1Let'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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