Chapter 7

Find the solutions of the equation that are in the interval \([0,2 \pi)\). $$ \cos t-\sin 2 t=0 $$

Exer. 1-38: Find all solutions of the equation. $$ \cos (\ln x)=0 $$

Exer. 1-50: Verify the identity. $$ \frac{1}{\tan \beta+\cot \beta}=\sin \beta \cos \beta $$

Exer. 37-46: Verify the identity. $$ \sin \left(\theta+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\sin \theta+\cos \theta) $$

Refer to Exercise 47 of Section 7.4. The graph of the equation \(y=\cos 3 x-3 \cos x\) has 13 turning points for \(-2 \pi \leq x \leq 2 \pi\). The \(x\)-coordinates of these points are solutions of the equation \(\sin 3 x-\sin x=0\). Use a sum-toproduct formula to find these \(x\)-coordinates.

Find the solutions of the equation that are in the interval \([0,2 \pi)\). $$ \cos u+\cos 2 u=0 $$

Exer. 33-42: Sketch the graph of the equation. $$ y=2 \cos ^{-1} x $$

Exer. 1-38: Find all solutions of the equation. $$ \ln (\sin x)=0 $$

Exer. 1-50: Verify the identity. $$ \frac{\cot y-\tan y}{\sin y \cos y}=\csc ^{2} y-\sec ^{2} y $$

Exer. 37-46: Verify the identity. $$ \cos \left(\theta+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\cos \theta-\sin \theta) $$

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

Exer. 33-42: Sketch the graph of the equation. $$ y=2+\tan ^{-1} x $$

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ \cos \left(2 x-\frac{\pi}{4}\right)=0 $$

Exer. 1-50: Verify the identity. $$ \sec \theta+\csc \theta-\cos \theta-\sin \theta=\sin \theta \tan \theta+\cos \theta \cot \theta $$

Exer. 37-46: Verify the identity. $$ \tan \left(u+\frac{\pi}{4}\right)=\frac{1+\tan u}{1-\tan u} $$

Vibration of a viotin string Mathematical analysis of a vibrating violin string of length \(I\) involves functions such that $$ f(x)=\sin \left(\frac{\pi n}{l} x\right) \cos \left(\frac{k \pi n}{l} t\right) $$ where \(n\) is an integer, \(k\) is a constant, and \(t\) is time. Express \(f\) as a sum of two sine functions.

Find the solutions of the equation that are in the interval \([0,2 \pi)\). $$ \tan 2 x=\tan x $$

Exer. 33-42: Sketch the graph of the equation. $$ y=\tan ^{-1} 2 x $$

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ \sin \left(3 x-\frac{\pi}{4}\right)=1 $$

Exer. 1-50: Verify the identity. $$ \sin ^{3} t+\cos ^{3} t=(1-\sin t \cos t)(\sin t+\cos t) $$

Exer. 37-46: Verify the identity. $$ \tan \left(x-\frac{\pi}{4}\right)=\frac{\tan x-1}{\tan x+1} $$

Pressure on the eardrum If two tuning forks are struck simultaneously with the same force and are then held at the same distance from the eardrum, the pressure on the outside of the eardrum at time \(t\) is given by $$ p(t)=a \cos \omega_{1} t+a \cos \omega_{2} t, $$ where \(a, \omega_{1}\), and \(\omega_{2}\) are constants. If \(\omega_{1}\) and \(\omega_{2}\) are almost equal, a tone is produced that alternates between loudness and virtual silence. This phenomenon is known as beats. (a) Use a sum-to-product formula to express \(p(t)\) as a product. (b) Show that \(p(t)\) may be considered as a cosine wave with approximate period \(2 \pi / \omega_{1}\) and variable amplitude \(f(t)=2 a \cos \frac{1}{2}\left(\omega_{1}-\omega_{2}\right) t\). Find the maximum amplitude. (c) Shown in the figure is a graph of the equation $$ p(t)=\cos 4.5 t+\cos 3.5 t . $$

Find the solutions of the equation that are in the interval \([0,2 \pi)\). $$ \tan 2 t-2 \cos t=0 $$

Exer. 33-42: Sketch the graph of the equation. $$ y=\sin (\arccos x) $$

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ 2-8 \cos ^{2} t=0 $$

Exer. 1-50: Verify the identity. $$ \left(1-\tan ^{2} \phi\right)^{2}=\sec ^{4} \phi-4 \tan ^{2} \phi $$

Exer. 37-46: Verify the identity. $$ \cos (u+v)+\cos (u-v)=2 \cos u \cos v $$

Find the solutions of the equation that are in the interval \([0,2 \pi)\). $$ \sin \frac{1}{2} u+\cos u=1 $$

Exer. 33-42: Sketch the graph of the equation. $$ y=\sin \left(\sin ^{-1} x\right) $$

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

Exer. 1-50: Verify the identity. $$ \cos ^{4} w+1-\sin ^{4} w=2 \cos ^{2} w $$

Exer. 37-46: Verify the identity. $$ \sin (u+v)+\sin (u-v)=2 \sin u \cos v $$

Find the solutions of the equation that are in the interval \([0,2 \pi)\). $$ 2-\cos ^{2} x=4 \sin ^{2} \frac{1}{2} x $$

Exer. 43-46: The given equation has the form \(y=f(x)\). (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Solve for \(x\) in terms of \(y\). $$ y=\frac{1}{2} \sin ^{-1}(x-3) $$

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

Exer. 1-50: Verify the identity. $$ \frac{\cot (-t)+\tan (-t)}{\cot t}=-\sec ^{2} t $$

Exer. 37-46: Verify the identity. $$ \sin (u+v) \cdot \sin (u-v)=\sin ^{2} u-\sin ^{2} v $$

If \(a>0, b>0\), and \(0

Exer. 43-46: The given equation has the form \(y=f(x)\). (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Solve for \(x\) in terms of \(y\). $$ y=3 \tan ^{-1}(2 x+1) $$

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

Exer. 1-50: Verify the identity. $$ \frac{\csc (-t)-\sin (-t)}{\sin (-t)}=\cot ^{2} t $$

Exer. 37-46: Verify the identity. $$ \cos (u+v) \cdot \cos (u-v)=\cos ^{2} u-\sin ^{2} v $$

Exer. 43-46: The given equation has the form \(y=f(x)\). (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Solve for \(x\) in terms of \(y\). $$ y=4 \cos ^{-1} \frac{2}{3} x $$

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

Exer. 1-50: Verify the identity. $$ \log 10^{\tan t}=\tan t $$

Exer. 37-46: Verify the identity. $$ \frac{1}{\cot \alpha-\cot \beta}=\frac{\sin \alpha \sin \beta}{\sin (\beta-\alpha)} $$

Exer. 43-46: The given equation has the form \(y=f(x)\). (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Solve for \(x\) in terms of \(y\). $$ y=2 \sin ^{-1}(3 x-4) $$

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

Exer. 1-50: Verify the identity. $$ 10^{\log |\sin t|}=|\sin t| $$

Exer. 37-46: Verify the identity. $$ \frac{1}{\tan \alpha+\tan \beta}=\frac{\cos \alpha \cos \beta}{\sin (\alpha+\beta)} $$