/6.4__Trigonometric_Functions

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6.4 – Trigonometric Functions  Now, we will consider an acute angle  in standard position. And choose a point P with coordinates  x , y  on the terminal side, and draw a right triangle. The side adjacent to  has length x and the side opposite  has length y. The length of the hypotenuse, r, is the distance from the origin, which can be found by

x2  y2  r 2 , so r  x2  y2 .

y hypotenuse opposite

  Now, the trigonometric ratios can be written in terms of x, y, and r. y x y sin  cos  tan  r r x x r r cot  sec  csc  y x y

x adjacent

EXAMPLE 1: Find the sine, cosine and tangent of the angle  , whose terminal side passes through 2, 1 . (HINT: First, you need to find r.)

 Let t be a real number such that it is an angle of t in radians in standard position. If any point  x , y  is on y r r csc t  y sint 

the terminal side of angle t, then:

x r r sec t  x

cos t 

y x x cot t  y

tant 

where r  x2  y2 is the distance from  x , y  to the origin.

x2  y2  1 , so now sint 

 In the UNIT CIRCLE, the radius 

y y x x   y and cos t    x . r 1 r 1

 Now, we have new Unit Circle descriptions of Trigonometric Functions: P has coordinates  cos t , sin t  tan t 

y sin t  x cos t

cot t 

x cos t  y sin t

sec t 

1 1  x cos t

csc t 

1 1  y sin t


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