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.

sohcahtoa

To which triangle(s) below does SOHCAHTOA apply?

SOCHAHTOA Video Lesson

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
sin(0°) = 0
30° sin(30°) = ½
90° sin(90°) = 1 (greatest value that sine can ever have)

What is the cosine ratio?

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
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

Problem 1

In the triangles below, identify the hypotenuse and the sides that are opposite and adjacent to the shaded angle.

Hypotenuse = AB
Opposite side = BC
Adjacent side = AC

Problem 2

Hypotenuse = AC
Opposite side = BC
Adjacent side = AB

Problem 3

Hypotenuse = YX
Opposite Side = ZX
Adjacent Side = ZY

Problem 4

Hypotenuse = I
Side opposite of A = H
Side adjacent to A = J

No Highlights (harder)

Problem 5

Identify the hypotenuse, and the opposite and adjacent sides of $$ \angle ACB $$

First, remember that the middle letter of the angle name( $$ \angle A\color{Red}{C}B $$) is the location of the angle.

Second:The key to solving this kind of problem is to remember that 'opposite' and 'adjacent' are relative to an angle of the triangle--which in this case is the red angle in the picture.

Problem 5

Identify the hypotenuse, and the opposite and adjacent sides of $$ \angle RPQ $$.

First, remember that the middle letter of the angle name($$ \angle R\color{Red}{P}Q $$) is the location of the angle.

Second:The key to solving this kind of problem is to remember that 'opposite' and 'adjacent' are relative to an angle of the triangle--which in this case is the red angle in the picture.

Problem 6

Identify the hypotenuse, and the opposite and adjacent sides of $$ \angle BAC $$.

First, remember that the middle letter of the angle name($$ \angle B\color{Red}{A}C $$) is the location of the angle.

Second:The key to solving this kind of problem is to remember that 'opposite' and 'adjacent' are relative to an angle of the triangle--which in this case is the red angle in the picture.

Problem 6

Identify the side that is opposite of $$\angle$$IHU and the side that is adjacent to $$\angle$$IHU.

First, remember that the middle letter of the angle name($$ \angle B\color{Red}{A}C $$) is the location of the angle.

Second:The key to solving this kind of problem is to remember that 'opposite' and 'adjacent' are relative to an angle of the triangle--which in this case is the red angle in the picture.

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