ࡱ> XZ WY -bjbj .% aaaaauuu8t!Lu'Km''''^(^(^(JJJJJJJ$LNVJa^(^(^(^(^(Jaa''JZJZJZJ^(b a'a'JZJ^(JZJZJZJ'bze5u3b ZJJJ0'KZJO"?8 OZJOaZJ8^(^(ZJ^(^(^(^(^(JJZJ^(^(^('K^(^(^(^(O^(^(^(^(^(^(^(^(^( ': LESSON 12 SUM AND DIFFERENCE FORMULAS The sum and difference formulas for the cosine function:  EMBED Equation.3   EMBED Equation.3  The sum and difference formulas for the sine function:  EMBED Equation.3   EMBED Equation.3  The sum and difference formulas for the tangent function:  EMBED Equation.3   EMBED Equation.3  Since  EMBED Equation.3 , then we can find the exact value of the cosine, sine, and tangent of  EMBED Equation.3  using the respective sum formula with  EMBED Equation.3  and  EMBED Equation.3 . Since  EMBED Equation.3  and  EMBED Equation.3 , then  EMBED Equation.3 . Of course, we could have obtained this by converting  EMBED Equation.3  to units of radians using the conversion factor  EMBED Equation.3 . That is  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3 . The cosine of  EMBED Equation.3  or  EMBED Equation.3 :  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  (a)  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  (b) The sine of  EMBED Equation.3  or  EMBED Equation.3 :  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  (c)  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  (d) The tangent of  EMBED Equation.3  or  EMBED Equation.3 :  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  (e)  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  (f) Since  EMBED Equation.3  is the reference angle for  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 , then we will be able to find the exact value of the six trigonometric functions for these angles. Since  EMBED Equation.3  is the reference angle for  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 , then we will be able to find the exact value of the six trigonometric functions for these angles. Examples Use a reference angle to find the exact value of the six trigonometric functions of the following angles. 1.  EMBED Equation.3  (This is the  EMBED Equation.3  angle in units of degrees.) The angle  EMBED Equation.3  is in the II quadrant. The reference angle of the angle  EMBED Equation.3  is the angle  EMBED Equation.3 . Since cosine is negative in the II quadrant and  EMBED Equation.3  =  EMBED Equation.3  by (b) above, then  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Since sine is positive in the II quadrant and  EMBED Equation.3  =  EMBED Equation.3  by (d) above, then  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Since tangent is negative in the II quadrant and  EMBED Equation.3  =  EMBED Equation.3  by (f) above, then  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  2.  EMBED Equation.3  (This is the  EMBED Equation.3  angle in units of radians.) The angle  EMBED Equation.3  is in the IV quadrant. The reference angle of the angle  EMBED Equation.3  is the angle  EMBED Equation.3 . Since cosine is positive in the IV quadrant and  EMBED Equation.3  =  EMBED Equation.3  by (a) above, then  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Since sine is negative in the IV quadrant and  EMBED Equation.3  =  EMBED Equation.3  by (c) above, then  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Since tangent is negative in the IV quadrant and  EMBED Equation.3  =  EMBED Equation.3  by (e) above, then  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  3.  EMBED Equation.3  (This is the  EMBED Equation.3  angle in units of radians.) The angle  EMBED Equation.3  is in the III quadrant. The reference angle of the angle  EMBED Equation.3  is the angle  EMBED Equation.3 . Since cosine is negative in the III quadrant and  EMBED Equation.3  =  EMBED Equation.3  by (a) above, then  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Since sine is negative in the III quadrant and  EMBED Equation.3  =  EMBED Equation.3  by (c) above, then  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Since tangent is positive in the III quadrant and  EMBED Equation.3  =  EMBED Equation.3  by (e) above, then  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Since  EMBED Equation.3 , then we can find the cosine, sine, and tangent of  EMBED Equation.3  using the respective difference formula with  EMBED Equation.3  and  EMBED Equation.3 . Since  EMBED Equation.3  and  EMBED Equation.3 , then  EMBED Equation.3 . Of course, we could have obtained this by converting  EMBED Equation.3  to units of radians using the conversion factor  EMBED Equation.3 . That is  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3 . The cosine of  EMBED Equation.3  or  EMBED Equation.3 :  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  (g)  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  (h) The sine of  EMBED Equation.3  or  EMBED Equation.3 :  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  (i)  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  (j) The tangent of  EMBED Equation.3  or  EMBED Equation.3 :  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  (k)  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  (l) NOTE: We also have that  EMBED Equation.3 . So, you can find the exact value of the cosine, sine, and tangent of  EMBED Equation.3  using the respective difference formula with  EMBED Equation.3  and  EMBED Equation.3 . Of course, you will obtain the same values that were obtained above. Since  EMBED Equation.3  is the reference angle for  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 , then we will be able to find the exact value of the six trigonometric functions for these angles. 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