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# 4 18 pts fill in the following table with the

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4. (18 pts.) Fill in the following table with the information requested concerning domain, range, and period. 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 . S o sin(2 θ ) = 2sin( θ )cos(
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4 18 pts Fill in the following table with the information...

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