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# Sine, Cosine and Tangent

## Opposite & adjacent sides and SOHCAHTOA of angles

This page explains the sine, cosine, tangent ratio, gives on an overview of their range of values and provides practice problems on identifying the sides that are opposite and adjacent to a given angle.
The Sine, Cosine and Tangent functions express the ratios of sides of a right triangle.

# To which triangle(s) below does SOHCAHTOA apply?

 SOHCAHTOA HOME SOHCAHTOA Worksheets Trigonometry Calculator

## What is the sine ratio?

Answer: sine of an angle is always the ratio of the $$\frac{opposite side}{hypotenuse}$$.
$sine(angle) = \frac{ \text{opposite side}}{\text{hypotenuse}}$

Example 1
 $$sin(\angle L) = \frac{opposite }{hypotenuse} = \frac{9}{15}$$
Example 2
 $$sin(\angle K) = \frac{opposite }{hypotenuse} = \frac{12}{15}$$
Remember: When we use the words 'opposite' and 'adjacent,' we always have to have a specific angle in mind.

Range of Values of Sine
For those comfortable in "Math Speak", the domain and range of Sine is as follows

Domain of Sine = all real numbers
Range of Sine = {-1 ≤ y ≤ 1}

The sine of an angle has a range of values from -1 to 1 inclusive. Below is a table of values illustrating some key sine values that span the entire range of values
 Angle Sine of the Angle 270° sin ( 270°) = -1 (smallest value that sine can ever have) 330° sin (330° ) = -½ 0° sin(0°) = 0 30° sin(30°) = ½ 90° sin(90°) = 1 (greatest value that sine can ever have)

## What is the cosine ratio?

Cosine Function
The cosine of an angle is always the ratio of the (adjacent side/ hypotenuse)
$cosine(angle) = \frac{ \text{adjacent side}}{\text{hypotenuse}}$
Example 1
 $$cos(\angle L) = \frac{adjacent }{hypotenuse} = \frac{12}{15}$$
Example 2
 $$cos(\angle K) = \frac{adjacent }{hypotenuse} = \frac{9}{15}$$

Range of Values of Cosine
For those comfortable in "Math Speak", the domain and range of cosine is as follows

Domain of Cosine = all real numbers
Range of Cosine = {-1 ≤ y ≤ 1}

The cosine of an angle has a range of values from -1 to 1 inclusive. Below is a table of values illustrating some key cosine values that span the entire range of values
 Angle Cosine of the Angle 0° cos (0°) = 1 ( greatest value that cosine can ever have) 60° cos (60° ) =½ 90° cos(90°) = 0 120° cos(120°) = -½ 180° cos(180°) = -1 (smallest value that sine can ever have)

## What is the tangent ratio?

The tangent of an angle is always the ratio of the (opposite side/ adjacent side)
$tangent(angle) = \frac{ \text{opposite side}}{\text{adjacent side}}$
Example 1
 $$tan(\angle L) = \frac{opposite }{adjacent } = \frac{9}{12}$$
Example 2
 $$tan(\angle K) = \frac{opposite }{adjacent } = \frac{12}{9}$$

Practice Problems

Highlighted Problems
In the triangles below, identify the hypotenuse and the sides that are opposite and adjacent to the shaded angle.
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Identify the hypotenuse, and the opposite and adjacent sides of $$\angle ACB$$.