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The Trigonometric Functions

This page is meant to serve as a quick overview of the basic trigonometric functions.

**Pythagorean Theorem**

The sum of the squares of the legs of a right triange equals the square of the hypotenuse, so * a² *+* b² *=* c² *.

**Distance Formula**

The Distance Formula is a variant of the Pythagorean Theorem that you use to calculate the distance between two points in a plane.

**Types of Angles**

An acute angle - An acute angle is an angle measuring between 0 and 90 degrees.

A right angle - A right angle is an angle measuring 90 degrees.

An obtuse angle - An obtuse angle is an angle measuring between 90 and 180 degrees.

A straight angle - A straight angle is an angle measuring 180 degrees.

Complementary Angles - Complementary angles are two angles whose sum of their degree measurements equals 90 degrees.

Supplementary Angles - Supplementary angles are two angles whose sum of their degree measurements equals 180 degrees.

Positive angles - An angle measurment that is measured counterclockwise from a given line.

Negative angles - An angle measurment that is measured clockwise from a given line.

Vertical Angles - A verticle angle is oppostive angles where any two lines that meet. Vertical angles have the same degree measurement.

Alternate Interior Angles - For any pair of parallel lines, that are both intersected by a third line, alternate interior angles are congruent.

Alternate Exterior Angles - For any pair of parallel lines, that are both intersected by a third line, alternate exterior angles are congruent.

**Basic Triangle Rules**

The sum of the measures of the angles of any triangle is 180 degrees.

Similar triangles have congruent corresponding angles and proporiaonal corresponding sides.

Triangles that are congruent are the same size and shape.

**Definition of Trigonometric Functions**

If (*x,y*) is any point other than the orgin on the terminal side of an angle Θ, and *r* = √(* x² *+* y² *), then the following applies.

sin Θ = y / r |
cos Θ = x / r |
tan Θ = y / x |

csc Θ = r / y |
sec Θ = r / x |
cot Θ = x / y |

Trigonometric Function Values for Quadrantal Angles | ||||||

Θ | sin Θ | cos Θ | tan Θ | cot Θ | sec Θ | csc Θ |

0° ( 0 ) | 0 | 1 | 0 | Undefined | 1 | Undefined |

90° ( π/2 ) | 1 | 0 | Undefined | 0 | Undefined | 1 |

180° ( π ) | 0 | -1 | 0 | Undefined | -1 | Undefined |

270°( 3π/2 ) | -1 | 0 | Undefined | 0 | Undefined | -1 |

360° ( 2π ) | 0 | 1 | 0 | Undefined | 1 | Undefined |

**Reciprocal Identities**

Reciprocal identities are trig identities that define cosecant, secant, and cotangent in terms of sine, cosine, and tangent and vise-versa.

sin Θ = 1 / csc Θ |
cos Θ = 1 / sec Θ |
tan Θ = 1 / cot Θ |

csc Θ = 1 / sin Θ |
sec Θ = 1 / cos Θ |
cot Θ = 1 / tan Θ |

**Pythagorean Identities**

Pythagorean Identities - Pythagorean identities are trig identities relating sine with cosine, tangent with secant, and cotangent with cosecant. These are all derived from the Pythagorean theorem.
* sin²Θ *+* cos²Θ *= 1
* tan²Θ *+ 1 = * sec²Θ *

1 + * cot²Θ *= * csc²Θ *