Chapter 6

Write the trigonometric expression as an algebraic expression. $$\sin (2 \arctan x)$$

Perform the addition or subtraction and simplify. $$\frac{1}{x+5}+\frac{x}{x-8}$$

Write the trigonometric expression as an algebraic expression. $$\cos (2 \arctan x)$$

The graph of a function \(f\) is shown over the 122, the graph of a function \(f\) is shown over the interval \([\mathbf{0}, \mathbf{2} \pi] .\) (a) Find the \(x\) -intercepts of the graph of \(f\) algebraically. Verify your solutions by using the zero or root feature of a graphing utility. (b) The \(x\) -coordinates of the extrema of \(f\) are solutions of the trigonometric equation. (Calculus is required to find the trigonometric equation.) Find the solutions of the equation algebraically. Verify these solutions using the maximum and minimum features of the graphing utility. Function: \(f(x)=\sin 2 x-\sin x\) Trigonometric Equation: \(2 \cos 2 x-\cos x=0\)

Perform the addition or subtraction and simplify. $$\frac{4 x}{x^{2}-25}-\frac{x^{2}}{x-5}$$

The graph of a function \(f\) is shown over the 122, the graph of a function \(f\) is shown over the interval \([\mathbf{0}, \mathbf{2} \pi] .\) (a) Find the \(x\) -intercepts of the graph of \(f\) algebraically. Verify your solutions by using the zero or root feature of a graphing utility. (b) The \(x\) -coordinates of the extrema of \(f\) are solutions of the trigonometric equation. (Calculus is required to find the trigonometric equation.) Find the solutions of the equation algebraically. Verify these solutions using the maximum and minimum features of the graphing utility. Function: \(f(x)=\cos 2 x+\sin x\) Trigonometric Equation: \(-2 \sin 2 x+\cos x=0\)

Perform the addition or subtraction and simplify. $$\frac{2 x}{x^{2}-4}+\frac{5}{x+4}$$

The range of a projectile fired at an angle \(\theta\) with the horizontal and with an initial velocity of \(v_{0}\) feet per second is given by $$r=\frac{1}{32} v_{0}^{2} \sin 2 \theta$$ where \(r\) is measured in feet. An athlete throws a javelin at 75 feet per second. At what angle must the athlete throw the javelin so that the javelin travels 130 feet?

Sketch the graph of the function. (Include two full periods.) $$f(x)=-\sin \pi x-1$$

The length of each of the two equal sides of an isosceles triangle is 10 meters (see figure). The angle between the two sides is \(\theta\) (a) Write the area of the triangle as a function of \(\theta / 2\). (b) Write the area of the triangle as a function of \(\theta\) and determine the value of \(\theta\) such that the area is a maximum.

Sketch the graph of the function. (Include two full periods.) $$f(x)=-2 \tan \frac{\pi x}{2}$$

Sketch the graph of the function. (Include two full periods.) $$f(x)=\frac{1}{2} \cot \left(x+\frac{\pi}{4}\right)$$

The Mach number \(M\) of an airplane is the ratio of its speed to the speed of sound. When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane (see figure). The Mach number is related to the apex angle \(\theta\) of the cone by $$\sin \frac{\theta}{2}=\frac{1}{M}$$ (a) Find the angle \(\theta\) that corresponds to a Mach number of 1. (b) Find the angle \(\theta\) that corresponds to a Mach number of 4.5 (c) The speed of sound is about 760 miles per hour. Determine the speed of an object having the Mach numbers in parts (a) and (b). (d) Rewrite the equation as a trigonometric function of \(\theta\).

Sketch the graph of the function. (Include two full periods.) $$f(x)=\cos (x-\pi)+3$$

Determine whether the statement is true or false. Justify your answer. $$\sin \frac{x}{2}=-\sqrt{\frac{1-\cos x}{2}}, \quad \pi \leq x \leq 2 \pi$$

Determine whether the statement is true or false. Justify your answer. The graph of \(y=4-8 \sin ^{2} x\) has a maximum at \((\pi, 4)\).

Consider the function \(f(x)=\sin ^{4} x+\cos ^{4} x\) (a) Use the power-reducing formulas to write the function in terms of cosine to the first power. (b) Determine another way of rewriting the function. Use a graphing utility to rule out incorrectly rewritten functions. (c) Add a trigonometric term to the function so that it becomes a perfect square trinomial. Rewrite the function as a perfect square trinomial minus the term that you added. Use the graphing utility to rule out incorrectly rewritten functions. (d) Rewrite the result of part (c) in terms of the sine of a double angle. Use the graphing utility to rule out incorrectly rewritten functions. (e) When you rewrite a trigonometric expression, your result may not be the same as a friend's. Does this mean that one of you is wrong? Explain.

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment connecting the points. $$(5,2),(-1,4)$$

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment connecting the points. $$(-4,-3),(6,10)$$

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment connecting the points. $$\left(-1, \frac{1}{2}\right),\left(\frac{4}{3}, \frac{5}{2}\right)$$

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment connecting the points. $$\left(\frac{1}{3}, \frac{2}{3}\right),\left(-1,-\frac{3}{2}\right)$$

Find (if possible) the complement and supplement of each angle. (a) \(55^{\circ}\) (b) \(162^{\circ}\)

Find (if possible) the complement and supplement of each angle. (a) \(109^{\circ}\) (b) \(78^{\circ}\)

Find (if possible) the complement and supplement of each angle. (a) \(\frac{\pi}{18}\) (b) \(\frac{9 \pi}{20}\)

Find (if possible) the complement and supplement of each angle. (a) \(\frac{2 \pi}{7}\) (b) \(\frac{11 \pi}{15}\)

Find the length of the arc on a circle of radius \(r\) intercepted by a central angle \(\theta\). $$r=21 \mathrm{cm}, \theta=35^{\circ}$$

Find the length of the arc on a circle of radius \(r\) intercepted by a central angle \(\theta\). $$r=15 \text { in. }, \theta-110^{\circ}$$