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# The Value of Special Angles of Trigonometric Functions

**Example 1**- The terminal side of angle a is pi/6. The radius of the circle is one, that is, r = 1.The P ( x , y ) is the intersection of the circle and the terminal side . Find the value of x and y, sin a , cos a , and tan a.

- Solution
- In right triangle OPM, since angle a = pi/6 = 30
^{o}, so the length of its opposite is one-half of the hypotenuse, that is, PM = r/2 , since r = 1 (given), so PM = 1/2. Since sin a = PM/r = PM/1 = PM, so PM is called the line of sin a. Since cos a = OM/r = OM/1 = OM, so OM is called the line of cos a. Since tan a = AT/OA = AT/1 = AT, so AT is called the line of tan a. (Note: OA = OP = r = 1 ).

**Example 2**- The terminal side of angle a is pi/4. Find the value of x and y, sin a , cos a , and tan a.

- Solution
- In right triangle POM, since angle a = pi/4 = 45
^{o}, so its both legs have the same length. That is, PM = OM. Since r = 1, so PM = OM = (square root 2)/2. PM is called the line of sin a. OM is called the line of cos a. AT is called the line of tan a. (Note: OA = OP = r = 1 ).

**Example 3**- The terminal side of angle a is pi/3. Find the value of x and y, sin a , cos a , and tan a.

- Solution
- In right triangle POM, since angle a = pi/3 = 60
^{o}, so angle OPM = 30^{o}, then the length of OM is one-half of the hypotenuse, that is, OM = r/2 , since r = 1 (given), so OM = 1/2. MP is called the line of sin a. OM is called the line of cos a. AT is called the line of tan a. (Note: OA = OP = r = 1 ).

**The Value of Special Angles of Sine, Cosine, and Tangent**- The following are the values of the special angles of sine, cosine, and tangent.

a in degree | a in radius | sin a | cos a | tan a |
---|---|---|---|---|

0^{o} |
0 | 0 | 1 | 0 |

30^{o} |
pi/6 | 1/2 | (square root of 3)/2 | (square root of 3)/3 |

45^{o} |
pi/4 | (square root of 2)/2 | (square root of 2)/2 | 1 |

60^{o} |
pi/3 | (square root of 3)/2 | 1/2 | square root of 3 |

90^{o} |
pi/2 | 1 | 0 | no exist |

180^{o} |
pi | 0 | -1 | 0 |

**Trigonometric Functions of Any Angles**- The following are the formula of trigonometric functions of any angles.

- k is a integer, k = ... -5,-4,-3,-2,-1,0,1,2,3,4,5 ...

**Example 4**- Find the value of sin 960
^{o}.

- Solution
- Formula used:
- sin(a + k × 360
^{o}) = sin(a) - sin(180
^{o}+ a) = - sin(a) - The terminal side of the angle a is in quadrant III, so sin(a) is < 0.

**Example 5**- Find the value of cos ( - 600
^{o}).

- Solution
- cos(-600
^{o}) - = cos(-2 × 360
^{o}+ 120^{o}) - = cos(120
^{o}) - = cos(180
^{o}- 60^{o}) - = - cos(60
^{o}) - = -1/2

- Formula used:
- cos(a + k × 360
^{o}) = cos a - cos(180
^{o}- a) = - cos a - The terminal side of the angle a is in quadrant II, so cos(a) is < 0.

**Example 6**- Find the value of tan 945
^{o}+ sin 510^{o}.

- Solution
- tan945
^{o}+ sin510^{o} - = tan(2 × 360
^{o}+ 225^{o}) + sin(1 × 360^{o}+ 150^{o}) - = tan225
^{o}+ sin150^{o} - = tan(180
^{o}+ 45^{}) + sin(180^{o}- 30^{o}) - = tan45
^{o}+ sin30^{o} - = 1 + 1/2
- = 3/2

- Formula used:
- tan(a + k × 360
^{o}) = tan a - In quadrant III, tan(a) > 0
- In quadrant II, sin(a) > 0

**Sine, Cosine, and Tangent of Special Angles**- The following are sine, cosine, and tangent of special angles whose terminal side coincides with x-axis or y-axis.

a in degree | 0^{o} |
90^{o} |
180^{o} |
270^{o} |
360^{o} |
---|---|---|---|---|---|

a in radius | 0 | pi/2 | pi | 3pi/2 | 2pi |

sin a | 0 | 1 | 0 | -1 | 0 |

cos a | 1 | 0 | -1 | 0 | 1 |

tan a | 0 | no exist | 0 | no exist | 0 |