Chapter 9

Perform each calculation. $$75^{\circ} 15^{\prime}+83^{\circ} 32^{\prime}$$

Graph each function over a two-period interval. Give the period and amplinde. $$y=\cos 2 x$$

Graph each function over a one-period interval. $$y=\sec (x-2 \pi)$$

Length of a Shadow Suppose that the angle of elevation of the sun is \(23.4^{\circ} .\) Find the length of the shadow cast by a woman who is 5.75 feet tall.

Consider the damped oscillatory function $$s(x)=5 e^{-0.3 x} \cos \pi x$$ (a) Graph \(y_{3}=5 e^{-0.3 x} \cos \pi x\) in \([0,3]\) by \([-5,5]\) (b) The graph of which function is the upper envelope of the graph of \(y_{3} ?\) (c) For what values of \(x\) does the graph of \(y_{3}\) touch the graph of the function found in part (b)?

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\csc \frac{\pi}{6}$$

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(1, \sqrt{3})$$

Perform each calculation. $$47^{\circ} 29^{\prime}-71^{\circ} 18^{\prime}$$

Graph each function over a two-period interval. Give the period and amplinde. $$y=\cos \frac{3}{4} x$$

Graph each function over a one-period interval. $$y=\csc (x+2 \pi)$$

Height of a Tower The shadow of a vertical tower is 40.6 meters long when the angle of elevation of the sun is \(34.6^{\circ} .\) Find the height of the tower.

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\sec \frac{\pi}{3}$$

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(-1, \sqrt{3})$$

Perform each calculation. $$47^{\circ} 23^{\prime}-73^{\circ} 48^{\prime}$$

Graph each function over a two-period interval. Give the period and amplinde. $$y=2 \sin \frac{1}{4} x$$

Graph each function over a one-period interval. $$y=3 \sec \frac{1}{4} x$$

From a window 30.0 feet above the street, the angle of elevation to the top of the building across the street is \(50.0^{\circ}\) and the angle of depression to the base of this building is \(20.0^{\circ} .\) Find the height of the building across the street.

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\csc \frac{\pi}{3}$$

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(\sqrt{2}, \sqrt{2})$$

Perform each calculation. $$90^{\circ}-72^{\circ} 58^{\prime} 11^{\prime \prime}$$

Graph each function over a two-period interval. Give the period and amplinde. $$y=3 \sin 2 x$$

Graph each function over a one-period interval. $$y=-2 \sec \frac{1}{2} x$$

The angle of elevation from the top of a small building to the top of a nearby taller building is \(46^{\circ} 40^{\prime},\) and the angle of depression to the bottom is \(14^{\circ} 10^{\prime} .\) If the smaller building is 28.0 meters high, find the height of the taller building.

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\cot \frac{\pi}{4}$$

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(-\sqrt{2},-\sqrt{2})$$

Graph each function over a two-period interval. Give the period and amplinde. $$y=-2 \cos 3 x$$

Graph each function over a one-period interval. $$y=-\frac{1}{2} \csc \left(x+\frac{\pi}{2}\right)$$

A student was asked to give the exact value of \(\sin 45^{\circ}\) Using his calculator, he gave the answer 0.7071067812 . The teacher did not give him credit. Why?

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(-2 \sqrt{3},-2)$$

Convert each angle measure to decimal degrees. Use a calculator, and round to the nearest thousandth of a degree if necessary. $$20^{\circ} 54^{\prime}$$

Graph each function over a two-period interval. Give the period and amplinde. $$y=-5 \cos 2 x$$

Graph each function over a one-period interval. $$y=\frac{1}{2} \csc \left(x-\frac{\pi}{2}\right)$$

A student was asked to give an approximate value of sin \(45 .\) With her calculator in degree mode, she gave the value \(0.7071067812 .\) The teacher did not give her credit. What was her error?

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(-2 \sqrt{3}, 2)$$

Convert each angle measure to decimal degrees. Use a calculator, and round to the nearest thousandth of a degree if necessary. $$38^{\circ} 42^{\prime}$$

Graph each function over a two-period interval. State the phase shift. $$y=\sin \left(x-\frac{\pi}{4}\right)$$

Graph each function over a one-period interval. $$y=\csc \left(x-\frac{\pi}{4}\right)$$

A tunnel is to be dug from \(A\) to \(B\). Both \(A\) and \(B\) are visible from \(C\). If \(A C\) is 1.4923 miles, \(B C\) is 1.0837 miles, and \(C\) is exactly \(90^{\circ}\) find the measures of angles \(A\) and \(B\)

Write each expression in terms of its co-function. $$\cot 73^{\circ}$$

An equation of the terminal side of an angle \(\theta\) in standard position is given with a restriction on \(x\). Sketch the least positive angle \(\theta\), and find the values of the six trigonometric functions of \(\theta\). $$2 x+y=0, x \geq 0$$

Convert each angle measure to decimal degrees. Use a calculator, and round to the nearest thousandth of a degree if necessary. $$91^{\circ} 35^{\prime} 54^{\circ}$$

Graph each function over a two-period interval. State the phase shift. $$y=\cos \left(x-\frac{\pi}{3}\right)$$

Graph each function over a one-period interval. $$y=\sec \left(x+\frac{3 \pi}{4}\right)$$

An airplane is fly. ing \(10,500\) feet above the level ground. The angle of depression from the plane to the base of a tree is \(13^{\circ} 50^{\prime}\) How far horizontally must the plane fly to be directly over the tree?

Write each expression in terms of its co-function. $$sec $39^{\circ}$$

An equation of the terminal side of an angle \(\theta\) in standard position is given with a restriction on \(x\). Sketch the least positive angle \(\theta\), and find the values of the six trigonometric functions of \(\theta\). $$3 x+5 y=0, x \geq 0$$

Convert each angle measure to decimal degrees. Use a calculator, and round to the nearest thousandth of a degree if necessary. $$34^{\circ} 51^{\prime} 35^{\prime \prime}$$

Graph each function over a two-period interval. State the phase shift. $$y=2 \cos \left(x-\frac{\pi}{3}\right)$$

Graph each function over a one-period interval. $$y=\sec \left(x+\frac{\pi}{4}\right)$$

The angle of elevation from a point on the ground to the top of a pyramid is \(35^{\circ} 30^{\prime} .\) The angle of elevation from a point 135 feet farther back to the top of the pyramid is \(21^{\circ} 10^{\prime} .\) Find the height of the pyramid.