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Searching for ## Trigonometric Ratios and Circles

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Geometry

Learners calculate sine, cosine, tangent, and cotangent for an acute angle.

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After completing this tutorial, you will be able to complete the following:

- Identify the opposite leg and adjacent leg of an acute angle in a right triangle.
- Identify the hypotenuse in a right triangle.
- Find the sine, cosine, tangent, and cotangent for an acute angle given in a right triangle in which all sides are given.
- Find the sine, cosine, tangent, and cotangent for an acute angle given in a right triangle in which only two sides are given.
- Find other trigonometric ratios of an acute angle when one ratio is given
- Find the sine, cosine, tangent, and cotangent for an acute angle given in a right triangle when one ratio of the other acute angle is given.

Basics of Trigonometry: Trigonometric Ratios in Right Triangles

The sound of a guitar string, the growth of a population, and the movement of an electron can all described with trigonometry. Trigonometry is the theory of triangles, which begins with a surprising fact about right triangles.

Right triangles, like the one shown above, have one degree angle and two acute angles (angles with measure less than 90 ). In the triangle above, one angle has been labeled (the Greek letter theta). Relative to , we say that side a is adjacent to and that side b is opposite .

The surprising fact about right triangles is that the ratios and are determined entirely by the angle . In other words, these ratios are a function of the angle . For example, for any 60 angle.

Trigonometric ratios: Sine, Cosine, and Tangent

Using the triangle above, the Sine, Cosine, and Tangent of are defined by the following equations.

The multiplicative inverse of tangent frequently occurs in applications, so it is helpful to have a separate function. The cotangent of is define by the following equation.

Using the definition of sine, along with , allows us to setup the following equation and solve for h :

This Activity Object focuses on answering such questions.

Approximate Time | 20 Minutes |

Pre-requisite Concepts | Angles in a right triangle, basics of the right triangle, Pythagorean theorem, rationalize the denominator of a fraction |

Course | Geometry |

Type of Tutorial | Concept Development |

Key Vocabulary | angles, cosine, cotangent |