91Ó°ÊÓ

Fill in the blanks. When solving a trigonometric equation, the preliminary goal is to ________ the trigonometric function involved in the equation.

Factor the trigonometric expression. There is more than one correct form of each answer. $$\sin ^{2} x+3 \cos x+3$$

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$75^{\circ}$$

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer. $$\frac{\tan \theta \cot \theta}{\sec \theta}$$

Use the sum-to-product formulas to rewrite the sum or difference as a product. $$\sin 5 \theta-\sin 3 \theta$$

Use the sum-to-product formulas to rewrite the sum or difference as a product. $$\cos 6 x+\cos 2 x$$

Rewrite the expression as a single logarithm and simplify the result. $$\ln |\cos x|-\ln |\sin x|$$

A weight is attached to a ,spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by $$y=\frac{1}{3} \sin 2 t+\frac{1}{4} \cos 2 t$$,where \(y\) is the distance from equilibrium (in feet) and \(t\) is the time (in seconds). (a) Use the identity \(a \sin B \theta+b \cos B \theta=\sqrt{a^{2}+b^{2}} \sin (B \theta+C)\) where \(C=\arctan (b / a), a>0,\) to write the model in the form \(y=\sqrt{a^{2}+b^{2}} \sin (B t+C)\). (b) Find the amplitude of the oscillations of the weight. (c) Find the frequency of the oscillations of the weight.

The height \(h\) (in feet) above ground of a seat on a Ferris wheel at time \(t\) (in minutes) can be modeled by $$h(t)=53+50 \sin \left(\frac{\pi}{16} t-\frac{\pi}{2}\right)$$. The wheel makes one revolution every 32 seconds. The ride begins when \(t=0\). (a) During the first 32 seconds of the ride, when will a person on the Ferris wheel be 53 feet above ground? (b) When will a person be at the top of the Ferris wheel for the first time during the ride? If the ride lasts 160 seconds, then how many times will a person be at the top of the ride, and at what times?

Find the smallest positive fixed point of the function \(f .\) A fixed point of a function \(f\) is a real number \(c\) such that \(f(c)=c\). $$f(x)=\tan (\pi x / 4)$$