Chapter 20

Use a graph to check that you have found all solutions in this interval. (Check \(f(x)=0.5\) on \([0,2 \pi]\) by graphing \(y=f(x)\) and \(y=0.5\) on \([0,2 \pi]\) and looking for points of intersection or by graphing \(y=f(x)-0.5\) on \([0,2 \pi]\) and looking for zeros. \()\) $$ 2 \sin (3 x)=-1 $$

In Problems 12 through 14, simplify the expressions given that \(x \in\left(\frac{\pi}{2}, 2 \pi\right) .\) (a) \(\arcsin (\sin x)\) (b) \(\arccos (\cos x)\)

Rewrite each of the following expressions in terms of a positive acute angle. This positive acute angle is sometimes referred to as a reference angle. (a) \(\cos 130^{\circ}\) (b) \(\cos \left(-130^{\circ}\right)\)

Due to the scampering of a goat, a rock has been dislodged from a mountain and s sliding down an incline making a \(70^{\circ}\) angle with level ground. The weight of the rock exerts a downward force of 3 pounds. What is the component of this force in the direction of motion of the rock?

Simplify the expressions given that \(x \in\left(\frac{\pi}{2}, 2 \pi\right) .\) (a) \(\arcsin (\sin (-x))\) (b) \(\arccos (\cos (-x))\)

Rewrite each of the following expressions in terms of a positive acute angle. This positive acute angle is sometimes referred to as a reference angle. (a) \(\cos 200^{\circ}\) (b) \(\sin \left(200^{\circ}\right)\)

Simplify the expressions given that \(x \in\left(\frac{\pi}{2}, 2 \pi\right) .\) a) \(\arctan (\tan (x))\) (b) \(\arctan (\tan (-x))\)

Rewrite each of the following expressions in terms of a positive acute angle. This positive acute angle is sometimes referred to as a reference angle. (a) \(\tan 200^{\circ}\) (b) \(\tan \left(-200^{\circ}\right)\)

Determine how many solutions each of the following equations has, and approximate the solutions. These equations involve both trigonometric and algebraic functions and therefore cannot be solved exactly using analytic methods. Instead, take a graphical approach. (a) \(\sin x-x=0\) (b) \(2 \cos x=x\) (c) \(\sin x-\frac{x}{3}=0\)

\text { Find all } t \in[0, \pi] \text { such that } 4 \sec ^{2}(2 t)-3=0 \text { . }

\text { Find all } t \in[-\pi, \pi] \text { such that } 2 \sin ^{2} t-3 \sin t+1=0 \text { . }

Find all \(x \in[0,2 \pi]\) such that $$ \cos 3 x=-\frac{1}{\sqrt{2}} $$

Find all solutions to $$ 5 \cos (x)=6 \cos ^{3}(x)-\sin (x) \cos (x) $$

\text { If } x \text { is a solution to } \cos x \sin x \cos (2 x) \sin (2 x)=0, \text { find } a l l \text { possible values of } \cos (x)

Find all values of \(x\) in the interval \([0,2 \pi]\) such that $$ \sin (4 x)=\frac{1}{\sqrt{2}} $$

A population of deer in a forest displays regular fluctuations in size. Scientists have chosen to model the population size with a sinusoidal function. At its height the deer population is 7000 , while at its low it is 2000 . The time between highs and lows is 6 months. The population at time \(t=0\) is 4500 and decreasing. (a) Model the deer population as a function of time \(t\) in months. A picture should accompany your answer. (b) If \(t=0\) is now, what is the deer population 3 months from now? (c) When is the first time in the future that the deer population will reach 3000 ? Give an exact answer and then a numerical approximation. (d) Call your answer to part (c) \(t_{*}\). Give any one time other than your answer to part (c) at which the deer population is also 3000 . (There are infinitely many correct answers.) Give an exact answer in terms of \(t_{*}\).