# Function name domain in radians range period in

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Function Name Domain (in radians) Range Period (in radians) sin( θ ) [-1,1] 2 π cos( θ ) [-1,1] 2 π tan( θ ) A, below π cot( θ ) B, below π sec( θ ) A, below (- ,-1] [1, ) 2 π csc( θ ) B, below (- ,-1] [1, ) 2 π A = { x ε : x (2k + 1)( π /2), k any integer } B = { x ε : x k π , k any integer }

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TEST-02/MAC1114 Page 2 of 3 5. (5 pts.) Establish the following identity. Show all steps very, very carefully. 1 - sin( α ) cos( α ) = cos( α ) 1 + sin( α ) Proof: 1 - sin( α ) [1 - sin( α )] [1 + sin( α )] = cos( α ) cos( α )[1 + sin( α )] 1 - sin 2 ( α ) = cos( α )[1 + sin( α )] cos 2 ( α ) = cos( α )[1 + sin( α )] cos( α ) = 1 + sin( α ) // 6. (5 pts.) Obtain the exact value of cos( π /8). Show clearly and neatly all the uses of appropriate identities. cos( π /8) = ([1 + cos(2( π /8))]/2) 1/2 = ([1 + cos(( π /4))]/2) 1/2 = ([1 + (2 1/2 /2)]/2) 1/2 = [(2 + 2 1/2 ) 1/2 ]/2 7. (5 pts.) If csc( θ ) = 4 and cos( θ ) < 0, what is the exact value of sin(2 θ ) ?? Show clearly and neatly all your uses of appropriate identities. sin( θ ) = 1/csc( θ ) = 1/4 and cos( θ ) = -(1 - sin 2 ( θ )) 1/2 = -15 1/2 /4 since cos( θ ) < 0. So sin(2 θ ) = 2sin( θ )cos( θ ) = 2 (1/4) (-15 1/2 /4) = -15 1/2 /8 8. (5 pts.) Express the following product as a sum containing only sines or cosines. sin(5 θ )cos(3 θ ) = (1/2)[sin(8 θ ) + sin(2 θ )] 9. (10 pts.) Find the exact value of each of the following expressions if tan( α ) = -5/12 with π /2 < α < π and sin( β ) = -1/2 with π < β < 3 π /2. Show all your uses of appropriate identities. sin( α ) = 5/13 and cos(a) = -12/13 since π /2 < α < π . cos( β ) = -3 1/2 /2 since π < β < 3 π /2.
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