91Ó°ÊÓ

Verify that each equation is an identity. $$\cos 2 \theta=\frac{2-\sec ^{2} \theta}{\sec ^{2} \theta}$$

Perform indicated operation and simplify the result. $$\tan s(\cot s+\csc s)$$

Solve each problem. Back Stress If a person bends at the waist with a straight back, making an angle of \(\theta\) degrees with the horizontal, then the force \(F\) exerted on the back muscles can be modeled by the equation $$F=\frac{0.6 W \sin \left(\theta+90^{\circ}\right)}{\sin 12^{\circ}}$$ where \(W\) is the weight of the person. (Source: Metcalf, H., Topics in Classical Biophysics, Prentice- Hall.) (a) Calculate \(F\) when \(W=170\) pounds and \(\theta=30^{\circ}\) (b) Use an identity to show that \(F\) is approximately equal to \(2.9 \mathrm{W} \cos \theta\) (c) For what value of \(\theta\) is \(F\) maximum?

Give solutions over the interval \([0,2 \pi)\) as approximations to the nearest hundredth when exact values cannot be determined. You may need to use the quadratic formula. Give approximate answers in Exercises \(59-64\) to the nearest tenth of a degree over the interval \(\left[0^{\circ}, 360^{\circ}\right)\) $$\cos ^{2} \theta+\sin \theta=0$$

Verify that each equation is an identity. $$\cot ^{2} \frac{x}{2}=\frac{(1+\cos x)^{2}}{\sin ^{2} x}$$

Give solutions over the interval \([0,2 \pi)\) as approximations to the nearest hundredth when exact values cannot be determined. You may need to use the quadratic formula. Give approximate answers in Exercises \(59-64\) to the nearest tenth of a degree over the interval \(\left[0^{\circ}, 360^{\circ}\right)\) $$\tan ^{2} \theta+2 \sec \theta-3=0$$

Draw by hand the graph of each inverse function. $$y=\cot ^{-1} 2 x$$

Perform indicated operation and simplify the result. $$\cos \beta(\sec \beta+\csc \beta)$$

Perform indicated operation and simplify the result. $$\frac{1}{\csc ^{2} \theta}+\frac{1}{\sec ^{2} \theta}$$

Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\cos \theta+1=0$$