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?

### SOCHAHTOA **Video** Lesson

### Interactive Angles**SOHCAHTOA**

**Status:**Angle activated : $$ \red{none} \text{, waiting for you to choose an angle.}$$

#### 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?

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

$$tan(\angle K) = \frac{opposite }{adjacent } = \frac{12}{9} $$