Chapter 7

Exer. 47-50: Solve the equation for \(x\) in terms of \(y\) if \(x\) is restricted to the given interval. $$ y=-3-\sin x ; \quad\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] $$

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ 2 \cos ^{2} \gamma+\cos \gamma=0 $$

Exer. 1-50: Verify the identity. $$ \ln \cot x=-\ln \tan x $$

Express \(\sin (u+v+w)\) in terms of trigonometric functions of \(u, v\), and \(w\). (Hint: Write \(\sin (u+v+w)\) as \(\sin [(u+v)+w]\) and use addition formulas.)

Exer. 47-50: Solve the equation for \(x\) in terms of \(y\) if \(x\) is restricted to the given interval. $$ y=2+3 \sin x ; \quad\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] $$

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ \sin x-\cos x=0 $$

Exer. 1-50: Verify the identity. $$ \ln \sec \theta=-\ln \cos \theta $$

Express \(\tan (u+v+w)\) in terms of trigonometric functions of \(u, v\), and \(w\).

Exer. 47-50: Solve the equation for \(x\) in terms of \(y\) if \(x\) is restricted to the given interval. $$ y=15-2 \cos x ; \quad[0, \pi] $$

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ \sin ^{2} \theta+\sin \theta-6=0 $$

Exer. 1-50: Verify the identity. $$ \ln |\sec \theta+\tan \theta|=-\ln |\sec \theta-\tan \theta| $$

Derive the formula \(\cot (u+v)=\frac{\cot u \cot v-1}{\cot u+\cot v}\).

Exer. 47-50: Solve the equation for \(x\) in terms of \(y\) if \(x\) is restricted to the given interval. $$ y=6-3 \cos x ; \quad[0, \pi] $$

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ 2 \sin ^{2} u+\sin u-6=0 $$

Exer. 1-50: Verify the identity. $$ \ln |\csc x-\cot x|=-\ln |\csc x+\cot x| $$

If \(\alpha\) and \(\beta\) are complementary angles, show that $$ \sin ^{2} \alpha+\sin ^{2} \beta=1 . $$

Projectile's range If a projectile is fired from ground level with an initial velocity of \(v \mathrm{ft} / \mathrm{sec}\) and at an angle of \(\theta\) degrees with the horizontal, the range \(R\) of the projectile is given by $$ R=\frac{v^{2}}{16} \sin \theta \cos \theta . $$ If \(v=80 \mathrm{ft} / \mathrm{sec}\), approximate the angles that result in a range of 150 feet.

Exer. 51-52: Solve the equation for \(x\) in terms of \(y\) if \(0

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ 1-\sin t=\sqrt{3} \cos t $$

Derive the subtraction formula for the sine function.

Exer. 51-52: Solve the equation for \(x\) in terms of \(y\) if \(0

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ \cos \theta-\sin \theta=1 $$

Exer. 51-60: Show that the equation is not an identity. (Hint: Find one number for which the equation is false.) $$ \sqrt{\sin ^{2} t+\cos ^{2} t}=\sin t+\cos t $$

Derive the subtraction formula for the tangent function.

Exer. 53-64: Use inverse trigonometric functions to find the solutions of the equation that are in the given interval, and approximate the solutions to four decimal places. \(\cos ^{2} x+2 \cos x-1=0\) \([0,2 \pi)\)

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ \cos \alpha+\sin \alpha=1 $$

Exer. 51-60: Show that the equation is not an identity. (Hint: Find one number for which the equation is false.) $$ \sqrt{\sin ^{2} t}=\sin t $$

If \(f(x)=\cos x\), show that $$ \frac{f(x+h)-f(x)}{h}=\cos x\left(\frac{\cos h-1}{h}\right)-\sin x\left(\frac{\sin h}{h}\right) $$

Exer. 53-64: Use inverse trigonometric functions to find the solutions of the equation that are in the given interval, and approximate the solutions to four decimal places. \(\sin ^{2} x-\sin x-1=0\) \([0,2 \pi)\)

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ \sqrt{3} \sin t+\cos t=1 $$

Exer. 51-60: Show that the equation is not an identity. (Hint: Find one number for which the equation is false.) $$ \sec t=\sqrt{\tan ^{2} t+1} $$

If \(f(x)=\tan x\), show that $$ \frac{f(x+h)-f(x)}{h}=\sec ^{2} x\left(\frac{\sin h}{h}\right) \frac{1}{\cos h-\sin h \tan x} . $$

AC circuit By definition, the average value of \(f(t)=c+a \cos b t\) for one or more complete cycles is \(c\) (see the figure). (a) Use a double-angle formula to find the average value of \(f(t)=\sin ^{2} \omega t\) for \(0 \leq t \leq 2 \pi / \omega\), with \(t\) in seconds. (b) In an electrical circuit with an alternating current \(I=I_{0} \sin \omega t\), the rate \(r\) (in calories/sec) at which heat is produced in an \(R\)-ohm resistor is given by \(r=R I^{2}\). Find the average rate at which heat is produced for one complete cycle.

Exer. 53-64: Use inverse trigonometric functions to find the solutions of the equation that are in the given interval, and approximate the solutions to four decimal places. \(2 \tan ^{2} t+9 \tan t+3=0 ; \quad\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ 2 \tan t-\sec ^{2} t=0 $$

Exer. 51-60: Show that the equation is not an identity. (Hint: Find one number for which the equation is false.) $$ (\sin \theta+\cos \theta)^{2}=\sin ^{2} \theta+\cos ^{2} \theta $$

Exer. 53-64: Use inverse trigonometric functions to find the solutions of the equation that are in the given interval, and approximate the solutions to four decimal places. \(3 \sin ^{2} t+7 \sin t+3=0 ; \quad\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ \tan \theta+\sec \theta=1 $$

Exer. 51-60: Show that the equation is not an identity. (Hint: Find one number for which the equation is false.) $$ \log \left(\frac{1}{\sin t}\right)=\frac{1}{\log \sin t} $$

Exer. 53-64: Use inverse trigonometric functions to find the solutions of the equation that are in the given interval, and approximate the solutions to four decimal places. \(15 \cos ^{4} x-14 \cos ^{2} x+3=0\); \([0, \pi]\)

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ \cot \alpha+\tan \alpha=\csc \alpha \sec \alpha $$

Exer. 51-60: Show that the equation is not an identity. (Hint: Find one number for which the equation is false.) $$ \cos (-t)=-\cos t $$

Exer. 57-62: Use an addition or subtraction formula to find the solutions of the equation that are in the interval \([0, \pi\) ). $$ \sin 4 t \cos t=\sin t \cos 4 t $$

Exer. 53-64: Use inverse trigonometric functions to find the solutions of the equation that are in the given interval, and approximate the solutions to four decimal places. \(3 \tan ^{4} \theta-19 \tan ^{2} \theta+2=0 ; \quad\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ \sin x+\cos x \cot x=\csc x $$

Exer. 57-62: Use an addition or subtraction formula to find the solutions of the equation that are in the interval \([0, \pi\) ). $$ \cos 5 t \cos 3 t=\frac{1}{2}+\sin (-5 t) \sin 3 t $$

Exer. 53-64: Use inverse trigonometric functions to find the solutions of the equation that are in the given interval, and approximate the solutions to four decimal places. \(6 \sin ^{3} \theta+18 \sin ^{2} \theta-5 \sin \theta-15=0 ; \quad\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ 2 \sin ^{3} x+\sin ^{2} x-2 \sin x-1=0 $$

Exer. 51-60: Show that the equation is not an identity. (Hint: Find one number for which the equation is false.) $$ \cos (\sec t)=1 $$

Exer. 57-62: Use an addition or subtraction formula to find the solutions of the equation that are in the interval \([0, \pi\) ). $$ \cos 5 t \cos 2 t=-\sin 5 t \sin 2 t $$