HW7sol - MAC1114 Homework 7 Due Tuesday March 1 1 Write the...

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Unformatted text preview: MAC1114 - Homework 7 Due Tuesday, March 1 1. Write the following in a form with no fractions: (a) 1 1 + cos x = 1 1 + cos x · 1- cos x 1- cos x = 1- cos x 1- cos 2 x = 1- cos x sin 2 x = 1 sin 2 x- cos x sin 2 x = csc 2 x- csc x cot x (b) cos 2 x 1- sec x = cos 2 x 1- sec x · 1 + sec x 1 + sec x = cos 2 x (1 + sec x ) 1- sec 2 x = cos 2 x (1 + sec x )- tan 2 x =- cot 2 x cos 2 x (1 + sec x ) 2. Write as a single logarithm and simplify: log 3 ( cos 2 x ) + log 3 ( sec 2 x + csc 2 x ) + log 3 ( sin 2 x ) = log 3 ( cos 2 x (sec 2 x + csc 2 x ) sin 2 x ) = log 3 ( cos 2 x 1 cos 2 x sin 2 x + cos 2 x 1 sin 2 x sin 2 x ) = log 3 ( sin 2 x + cos 2 x ) = log 3 1 = 0 3. Use the trigonometric substitution x = 2 3 tan θ , for < θ < π 2 , to rewrite the equation √ 9 x 2 + 4 = 3 as a function of θ , and nd cos θ (you will need to draw a triangle). Actually, you don't need a triangle to nd cos θ , but you would to nd any of the other trig functions....
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