Sum and Difference Formulas At this point, you should know how to find the frig values of common angles like 6: 7: 7: and quadrantal angles like 0, 2 .. without using a calculator. The goal now is to combine these trig values to find new trig values by using the sum and difference formulas: sin(a + B) = sin(a) cos(B) + cos(a) sin(B) sin(a - P) = sin(a) cos(B) - cos(a) sin(B) cos(a + 8) = cos(a) cos(B) - sin(a) sin(B) cos(a - B) = cos(c) cos(B) + sin(c) sin(B) tan(a) + tan(B) tan(a + 8) = 1 - tan(o) tan(B) For example, let's consider the angles o = * and B =- and find their sum and difference: 6 6 - = 57 6 We can find the trig values of - 7 7 6 and 6 - by using the trig values of a and - into the sum and difference formulas: COS = cos (+ ) = cos(IT) cos - sin(7) sin (= ) = (-1) . V3 - 0 V3 6 2 ons (a+ 8) cos(a + 8) cos () cos (8) sin() sin (8) sin 6 - sin (*+6) = sin(7) cos () + cos(7) sin ( ) =0. V3 N/ H 2 + (-1 ) . - = sin(at #) sin(a + 8) sin (a) cos(#) cos(a) sin(P) 57T COS 6 = COS ( ST - = cos (7) cos () + sin() sin () = (-1) . V3 V3 2 +0 2 cos(a-8) COS (CE) COB(B) sin(c) sin (8) 57T sin = sin (x -6 ) = sin() cos () =0. V3 6 - cos(7) sin () 2 (-1) . 5 = sin (a- 8) sin(a-8) sin(a) cos(8) cos(a) sin(P). Decimal approximations are not allowed for this problem. . Enter your answer in exact form. . Use \"sqrh: )' to represent VI. 23w Use a sum or clierence formula to compute the exact value of tan( )_ \"4%) =8 Double and Half Angle Formulas Recall the double angle formulas: sin(2c2] = 25in(a) coda) 605(20] cosz(a) 51:11.2(0) 1 2sin2(a) 2c052(a) 1 2tan(a] 1 taught) tan(2c2] = Recall the half angle formulas: a: _ __ 1 cos(a] 5]\" (E) _ \"'ll 2 a: _ __ 1 -- cos(a] ....(2)___ 2 a: 1 cos(a] t = :: an ( 2 ) 1 -- cos(a] Example: Sincecos(] tth ' I ' - 4 2 , ge e cosine of 3 via the hartangle formula. 7r _ \"/4 ___ 1+m(1)_ f1+ 006(3) m( 2) " 2 4 22 2 . J2 1 2 + J2 2 + J2 _ 2 ' 2 2 _ 4 _ 2 Here, we used the '+' sign, because the angle g = g = 22.5\" is in the rst quadrant, so that it's cosine is positive. Practice Use the half angle formulas to nd the following trigonometric function values. - Decimal approximations are not allowed for this problem. - Enter your answer in exact form. - Use "sqrt( )\" to represent '/ Trigonometry - Double and Half Angle Formulas: Problem 2 [1 point} 1. Given that a is in Quadrant 4 and 005(01) = %, give an exact answer for the following: a. siJ1{2a) =C] b. cos(2a) =:] c. tan(2a] =:] 2. Given that ,8 is in Quadrant 4 and tanm) = 'Ts, give an exact answer for the following: b. 1305(23) =:] c. tan(2,6) =:] - Decimal approximations are not allowed for this problem. - Enter your answer in exact form. - Use "sqrt( )\" to represent \\/