Chapter 13

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. If the treadmill bed is 40 inches long, what is the vertical rise when set at an 8% incline?

Find the exact value of each trigonometric function. $$ \cot \left(\frac{\pi}{3}\right) $$

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 \(\tan \theta-m\) Determine the obtuse angle formed at the intersection of the graphs of \(2 x+5 y=8\) and \(6 x-y=-8 .\) State the measure of the angle to the nearest degree.

Solve each equation or inequality. \(4 e^{x}-3>-1\)

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 \(\theta .\) \(\tan \theta=-\frac{1}{5},\) Quadrant \(\Pi\)

Rewrite each degree measure in radians and each radian measure in degrees. \(260^{\circ}\)

Find the exact value of each trigonometric function. $$ \csc \left(\frac{\pi}{4}\right) $$

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 \(\tan \theta-m\) Explain why this relationship, \(\tan \theta=m,\) holds true.

Solve each equation or inequality. \(\ln (x+3)=2\)

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 \(\theta .\) \(\sin \theta=\frac{1}{3}\) Quadrant II

Rewrite each degree measure in radians and each radian measure in degrees. \(\frac{29 \pi}{4}\)

A geologist measured a \(40^{\circ}\) of elevation to the top of a mountain. After moving 0.5 kilometer farther away, the angle of elevation was \(34^{\circ} .\) How high is the top of the mountain? (Hint: Write a system of equations in two variables.)

Find one angle with positive measure and one angle with negative measure coterminal with each angle. $$ 300^{\circ} $$

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 \(\theta .\) \(-\frac{2 \pi}{3}\)

Rewrite each degree measure in radians and each radian measure in degrees. \(\frac{17 \pi}{6}\)

Draw two right triangles \(\triangle A B C\) and \(\triangle D E F\) for which sin \(A=\) sin \(D\) . What can you conclude about \(\triangle A B C\) and \(\triangle D E F\) ? Justify your reasoning.

Find one angle with positive measure and one angle with negative measure coterminal with each angle. $$ 47^{\circ} $$

Find a counterexample to the statement It is always possible to solve a right triangle.

Find one angle with positive measure and one angle with negative measure coterminal with each angle. $$ \frac{5 \pi}{3} $$

Find the area of \(\triangle A B C\) . Round to the nearest tenth. $$ a=11 \text { in. } c=5 \text { in. }, B=79^{\circ} $$

Explain why the sine and cosine of an acute angle are never greater that 1, but the tangent of an acute angle may be greater than 1.

A rocket rises 20 feet in the first second, 60 feet in the second second, and 100 feet in the third second. If it continues at this rate, how many feet will it rise in the 20 th second?

Find the exact value of each function. \(\sin -660^{\circ}\)

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(\frac{\pi}{2}\)

Find the area of \(\triangle A B C\) . Round to the nearest tenth. $$ b=4 \mathrm{m}, c=7 \mathrm{m}, A=63^{\circ} $$

CAROUSELS Anthony's little brother gets on a carousel that is 8 meters in diameter. At the start of the ride, his brother is 3 meters from the fence to the ride. How far will his brother be from the fence after the carousel rotates \(240^{\circ} ?\)

Solve each equation. Round to the nearest tenth. $$ a^{2}=3^{2}+5^{2}-2(3)(5) \cos 85^{\circ} $$

Find the exact value of each function. \(\cos 25 \pi\)

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

Find the sum of each infinite geometric series, if it exists. $$ a_{1}=3, r=1.2 $$

sKYCOASTING Mikhail and Anya visit a local amusement park to ride a skycoaster. After the first several swings, the angle the skycoaster makes with the vertical is modeled by \(\theta=0.2 \mathrm{cos} \pi t,\) with \(\theta\) measured in radians and \(t\) measured in seconds. Determine the measure of the angle for \(t=0,\) \(0.5,1,1.5,2,2.5,\) and 3 in both radians and degrees.

If the secant of angle \(\theta\) is \(\frac{25}{7}\) , what is the sine of angle \(\theta ?\) A. \(\frac{5}{25}\) B. \(\frac{7}{25}\) C. \(\frac{24}{25}\) D. \(\frac{25}{7}\)

Solve each equation. Round to the nearest tenth. $$ c^{2}=12^{2}+10^{2}-2(12)(10) \cos 40^{\circ} $$

Find the exact value of each function. \(\left(\sin 135^{\circ}\right)^{2}+\left(\cos -675^{\circ}\right)^{2}\)

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(\frac{4 \pi}{3}\)

Find the sum of each infinite geometric series, if it exists. $$ 16,4,1, \frac{1}{4}, \ldots $$

NAVIGATION Ships and airplanes measure distance in nautical miles. The formula 1 nautical mile \(=6077-31\) cos 2\(\theta\) feet, where \(\theta\) is the latitude in degrees, can be used to find the approximate length of a nautical mile at a cegrees, can be used to find the approximate length of a nautical mile at a certain latitude. Find the length of a nautical mile where the latitude is \(60^{\circ} .\)

A person holds one end of a rope that runs through a pulley and has a weight attached to the other end. Assume the weight is directly beneath the pulley. The section of rope between the pulley and the weight is 12 feet long. The rope bends through an angle of 33 degrees in the pulley. How far is the person from the weight? F 7.8 ft G 10.5 ft H 12.9 ft J 14.3 ft

Solve each equation. Round to the nearest tenth. $$ 7^{2}=11^{2}+9^{2}-2(11)(9) \cos B^{\circ} $$

Determine whether each triangle should be solved by beginning with the Law of Sines or Law of Cosines. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. $$ a=3.1, b=5.8, A=30^{\circ} $$

Find the sum of each infinite geometric series, if it exists. $$ \sum_{n=1} 13(-0.625)^{n-1} $$

OPEN ENDED Give an example of an angle for which the sine is negative and the tangent is positive.

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(-\frac{2 \pi}{3}\)

Determine whether each situation would produce a random sample. Write yes or no and explain your answer surveying band members to find the most popular type of music at your school

Solve each equation. Round to the nearest tenth. $$ 13^{2}=8^{2}+6^{2}-2(8)(6) \cos A^{\circ} $$

Determine whether each triangle should be solved by beginning with the Law of Sines or Law of Cosines. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. $$ a=9, b=40, c=41 $$

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(\frac{9 \pi}{2}\)

Determine whether each situation would produce a random sample. Write yes or no and explain your answer surveying people coming into a post office to find out what color cars are most popular

Use synthetic substitution to find \(f(3)\) and \(f(-4)\) for each function. $$ f(x)=5 x^{2}+6 x-17 $$

PREREQUISITE SKILL Find each value of \(\theta\) . Round to the nearest degree. $$ \cos \theta=-0.3420 $$