Trigonometry

Study of triangles and trigonometric functions including sine, cosine, and tangent.

Understanding Angles and TrianglesSine, Cosine, and Tangent FunctionsThe Unit Circle and Radian Measure
Trigonometric Identities and FormulasSolving TrianglesGraphs of Trigonometric Functions
Measuring Heights and DistancesTrigonometry in Engineering and DesignNavigation and GPS
Memorizing Trigonometric ValuesSolving Word Problems with DiagramsChecking Your Work with Inverse Functions
Understanding Angles...Sine, Cosine, and Ta...The Unit Circle and ...Trigonometric Identi...Solving TrianglesGraphs of Trigonomet...Measuring Heights an...Trigonometry in Engi...Navigation and GPSMemorizing Trigonome...Solving Word Problem...Checking Your Work w...
Basic Concepts

Sine, Cosine, and Tangent Functions

Introducing the Trigonometric Functions

Trigonometry is all about the relationships between the sides and angles of triangles. The three most important functions are sine, cosine, and tangent.

Definitions

In a right triangle:

  • Sine (\(\sin\)): \( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
  • Cosine (\(\cos\)): \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
  • Tangent (\(\tan\)): \( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \)

These functions help you find missing sides or angles in right triangles.

Why are they useful?

They can turn angle measurements into side lengths, and vice versa!

Key Formula

\[\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}\]

Examples

  • If the opposite side is 4 and the hypotenuse is 5, then \( \sin(\theta) = \frac{4}{5} \).

  • If the adjacent side is 3 and the hypotenuse is 5, then \( \cos(\theta) = \frac{3}{5} \).

In a Nutshell

Sine, cosine, and tangent connect angles to side lengths in right triangles.

Key Terms

Sine
Ratio of the length of the side opposite an angle to the hypotenuse.
Cosine
Ratio of the length of the adjacent side to the hypotenuse.
Tangent
Ratio of the length of the side opposite an angle to the adjacent side.
Next
The Unit Circle and Radian Measure