Chapter 13

Find each value. Write angle measures in radians. Round to the nearest hundredth. \(\cos \left(\cos ^{-1} \frac{\sqrt{2}}{2}-\frac{\pi}{2}\right)\)

Mateo and Amy are deciding which method, the Law of Sines or the Law of Cosines, should be used first to solve \(\triangle A B C\) . Mateo Begin by using the Law of Sines, since you are given two sides and an angle opposite one of them. Amy Begin by using the Law of Cosines, since you are given two sides and their included angle. Who is correct? Explain your reasoning.

BIOLOCY In a certain area of forested land, the population of rabbits \(R\) increases and decreases periodically throughout the year. If the population can be modeled by \(R=425+200 \sin \left[\frac{\pi}{365}(d-60)\right],\) where \(d\) represents the \(d\) th day of the year, describe what happens to the population throughout the year.

Find the exact value of each trigonometric function. \(\cos \left(-30^{\circ}\right)\)

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(\frac{7 \pi}{6}\)

Solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. $$ A=50^{\circ}, a=2.5, c=3 $$

Find the exact value of each trigonometric function. \(\tan \left(-\frac{5 \pi}{4}\right)\)

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(-\frac{5 \pi}{4}\)

Solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. $$ B=18^{\circ}, C=142^{\circ}, b=20 $$

Find each value. Write angle measures in radians. Round to the nearest hundredth. \(\sin \left(2 \cos ^{-1} \frac{3}{5}\right)\)

Explain how the Pythagorean Theorem is a special case of the Law of Cosines.

Sketch each angle. Then find its reference angle. \(315^{\circ}\)

A sector is a region of a circle that is bounded by a central angle \(\theta\) and its intercepted arc. The area \(A\) of a sector with radius \(r\) and central angle \(\theta\) is given by \(A=\frac{1}{2} r^{2} \theta,\) where \(\theta\) is measured in radians. Find the area of a sector with a central angle of \(\frac{4 \pi}{3}\) radians in a circle whose radius measures 10 inches.

BALLOONING As a hot-air balloon crosses over a straight portion of interstate highway, its pilot eyes two consecutive mileposts on the same side of the balloon. When viewing the mileposts the angles of depression are \(64^{\circ}\) and \(7^{\circ} .\) How high is the balloon to the nearest foot?

Find each value. Write angle measures in radians. Round to the nearest hundredth. \(\sin \left(2 \sin ^{-1} \frac{1}{2}\right)\)

Sketch each angle. Then find its reference angle. \(240^{\circ}\)

OPEN ENDED Give an example of a triangle that has two solutions by listing measures for \(A, a,\) and \(b,\) where a and \(b\) are in centimeters. Then draw both cases using a ruler and protractor.

Fountains Architects who design fountains know that both the height and distance that a water jet will project is dependent on the angle \(\theta\) at which the water is aimed. For a given angle \(\theta\) , the ratio of the maximum height \(H\) of the parabolic arc to the horizontal distance \(D\) it travels is given by \(\frac{H}{D}=\frac{1}{4} \tan \theta .\) Find the value of \(\theta,\) to the nearest degree, that will cause the arc to go twice as high as it travels horizontally.

Sketch each angle. Then find its reference angle. \(\frac{5 \pi}{4}\)

Draw an angle with the given measure in standard position. \(-150^{\circ}\)

TRACK AND FIELD A shot put must land in a \(40^{\circ}\) sector. nThe vertex of the sector is at the origin and one side lies along the \(x\) -axis. An athlete puts the shot at a point with coordinates \((18,17),\) did the shot land in the required region? Explain your reasoning.

Two trucks, A and B, start from the intersection C of two straight roads at the same time. Truck A is traveling twice as fast as truck B and after 4 hours, the two trucks are 350 miles apart. Find the approximate speed of truck B in miles per hour. F. 35 G. 37 H. 57 J. 73

Sketch each angle. Then find its reference angle. \(\frac{5 \pi}{6}\)

Draw an angle with the given measure in standard position. \(-50^{\circ}\)

Solve \(\triangle A B C\) by using the given measurements. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. \(\tan B=\frac{7}{8}, b=7\)

REASONING Determine whether the following statement is sometimes, always or never true. Explain your reasoning. If given the measure of two sides of a triangle and the angle opposite tone of them, you will be able to find a unique solution.

For Exercises \(38-40,\) consider \(f(x)=\sin ^{-1} x+\cos ^{-1} x\) Make a table of values, recording \(x\) and \(f(x)\) for \(x=\left\\{0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1,-\frac{1}{2},\right.\) \(-\frac{\sqrt{2}}{2},-\frac{\sqrt{3}}{2},-1 \\}\)

Mr. Blackwell is building a triangular sandbox. He is to join a 3-meter beam to a 4 meter beam so the angle opposite the 4-meter beam measures \(80^{\circ} .\) To what length should Mr. Blackwell cut the third beam in order to form the triangular sandbox? Round to the nearest tenth.

Sketch each angle. Then find its reference angle. \(-210^{\circ}\)

Draw an angle with the given measure in standard position. \(\pi\)

For Exercises \(38-40,\) consider \(f(x)=\sin ^{-1} x+\cos ^{-1} x\) Make a conjecture about \(f(x)\)

Find the exact values of the six trigonometric functions of \(\theta\) if the terminal side of \(\theta\) in standard position contains the given point. \((5,12)\)

OPEN ENDED Give an example of a situation that could be described by a periodic function. Then state the period of the function.

Sketch each angle. Then find its reference angle. \(-125^{\circ}\)

Draw an angle with the given measure in standard position. \(-\frac{2 \pi}{3}\)

Find the exact values of the six trigonometric functions of \(\theta\) if the terminal side of \(\theta\) in standard position contains the given point. \((4,7)\)

Sketch each angle. Then find its reference angle. \(\frac{13 \pi}{7}\)

Rewrite each degree measure in radians and each radian measure in degrees. \(660^{\circ}\)

REVIEW The longest side of a triangle is 67 inches. Two angles have measures of \(47^{\circ}\) and \(55^{\circ} .\) What is the length of the shortest leg of the triangle? $$ \begin{array}{ll}{\text { F } 50.1 \text { in. }} & {\text { H } 60.1 \text { in. }} \\ {\text { G } 56.1 \text { in. }} & {\text { J } 62.3 \text { in. }}\end{array} $$

OPEN ENDED Write an equation giving the value of the Cosine function for an angle measure in its domain. Then, write your equation in the form of an inverse function.

Find the exact values of the six trigonometric functions of \(\theta\) if the terminal side of \(\theta\) in standard position contains the given point. \((\sqrt{10}, \sqrt{6})\)

CHALLENGE Determine the domain and range of the functions \(y=\sin \theta\) and \(y=\cos \theta\)

Sketch each angle. Then find its reference angle. \(-\frac{2 \pi}{3}\)

Rewrite each degree measure in radians and each radian measure in degrees. \(570^{\circ}\)

A preprogrammed workout on a treadmill consists of intervals walking at various rates and angles of incline. A 1% incline means 1 unit of vertical rise for every 100 units of horizontal run. At what angle, with respect to the horizontal, is the treadmill bed when set at a 10% incline? Round to the nearest degree.

Find the exact value of each trigonometric function. $$ \cos 30^{\circ} $$

CHALLENGE For Exercises \(42-44,\) use the following information. If the graph of the line \(y=m x+b\) intersects the \(x\) -axis such that an angle of \(\theta\) is formed with the positive \(x\) -axis, then \(\tan \theta-m\) Find the acute angle that the graph of \(3 x+5 y=7\) makes with the positive \(x\) -axis to the nearest degree.

Solve each equation or inequality. \(e^{x}+5=9\)

Suppose \(\theta\) is an angle in standard position whose terminal side is in the given quadrant. For each function, find the exact values of the remaining five trigonometric functions of \(\theta .\) \(\cos \theta=\frac{3}{5},\) Quadrant IV

Rewrite each degree measure in radians and each radian measure in degrees. \(158^{\circ}\)