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Acute Angles and Right Triangles

This page is meant to serve as a quick overview of the basics of acute angles and right triangles.

**Right Triangle Based Definitions of the Trigonometric Functions**

For any acute angle Θ in standard position, the following applies:

sin Θ = y / r = (opposite side) / (hypotenuse) |
cos Θ = x / r = (adjacent side) / (hypotenuse) |
tan Θ = y / x = (opposite side) / (adjacent side) |

csc Θ = r / y = (hypotenuse) / (opposite side) |
sec Θ = r / x = (hypotenuse) / (adjacent side) |
cot Θ = x / y = (adjacent side) / (opposite side) |

**Cofunction Identities**

For any acute angle Θ in standard position, the following applies:

sin Θ = cos(90-Θ) | cos Θ = sin(90-Θ) | tan Θ = cot(90-Θ) |

csc Θ = sec(90-Θ) | sec Θ = csc(90-Θ) | cot Θ = tan(90-Θ) |

Function Valuse of Special Angles | ||||||

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

30° ( π/6 ) | 1 / 2 | √3 / 2 | √3 / 3 | √3 | 2√3 / 3 | 2 |

45° ( π/4 ) | √2 / 2 | √2 / 2 | 1 | 1 | √2 | √2 |

60° ( π/3 ) | √3 / 2 | 1 / 2 | √3 | √3 / 3 | 2 | 2√3 / 3 |

**Reference Angle Θ' for Θ in (0°,360°)**

Reference Angle - A Reference Angle an acute version of any given angle. In the standard position, the reference angle is the smallest angle between the terminal side and the x-axis. The *absolute value* of the trig functions of angle *Θ* are the same as the trig values of the reference angle for *Θ*.

Θ in Quadrant |
Θ' is |

I | Θ |

II | 180° Θ |

III | Θ - 180° |

IV | 360° - Θ |

**Procedures for Discovering the Trigonometric Values for any Angle**

Step 1 | Add or subtract 360° as many times as necessary to get the angle between 0° and 360°. |

Step 2 | Find the reference angle using the chart above. |

Step 3 | Calculate the trigonometric values for Θ'. |

Step 4 | Dtermine the proper sign for the value found in step 3 using ASTC (All Sin Tangent Cos). |

**Solving Applied Trigonometry Problems**

Step 1 | Sketch a diagram and lable it with all the givens. Provide the quantity to be found with a variable. |

Step 2 | Write an equation based on the sketch from Step 1 that relates the variable with the given quantities. |

Step 3 | Solve for the variable. Check the diagram and ensure the answer makes sense. |