삼각함수의 뜻과 관계

For the following exercises, use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think $2x=x+x.$ )

$f(\theta )=\cos (2\theta ),g(\theta )={\cos }^{2}\theta -{\sin }^{2}\theta$

For the following exercises, use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think $2x=x+x.$ )

$f(\theta )=\tan (2\theta ),g(\theta )=\frac{\tan \,\theta }{1+{\tan }^{2}\theta }$

For the following exercises, use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think $2x=x+x.$ )

$f(x)=\sin (3x)\sin \,x,g(x)={\sin }^{2}(2x){\cos }^{2}x-{\cos }^{2}(2x){\sin }^{2}x$

For the following exercises, use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think $2x=x+x.$ )

$f(x)=\tan (-x),g(x)=\frac{\tan \,x-\tan (2x)}{1-\tan \,x\,\tan (2x)}$

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.

$\sin (75^\circ )$

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.

$\sin (195^\circ )$

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.

$\cos (165^\circ )$

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.

$\cos (345^\circ )$

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.

$\tan (-15^\circ )$

For the following exercises, prove the identities provided.

$\tan \left(x+\frac{\pi }{4}\right)=\frac{\tan \,x+1}{1-\tan \,x}$

For the following exercises, prove the identities provided.

$\frac{\tan (a+b)}{\tan (a-b)}=\frac{\sin \,a\,\cos \,a+\sin \,b\,\cos \,b}{\sin \,a\,\cos \,a-\sin \,b\,\cos \,b}$

For the following exercises, prove the identities provided.

$\frac{\cos (a+b)}{\cos \,a\,\cos \,b}=1-\tan \,a\,\tan \,b$

For the following exercises, prove the identities provided.

$\cos (x+y)\cos (x-y)={\cos }^{2}x-{\sin }^{2}y$

For the following exercises, prove the identities provided.

$\frac{\cos (x+h)-\cos \,x}{h}=\cos \,x\frac{\cos \,h-1}{h}-\sin \,x\frac{\sin \,h}{h}$

For the following exercises, prove or disprove the statements.

$\tan (u+v)=\frac{\tan \,u+\tan \,v}{1-\tan \,u\,\tan \,v}$

For the following exercises, prove or disprove the statements.

$\tan (u-v)=\frac{\tan \,u-\tan \,v}{1+\tan \,u\,\tan \,v}$

For the following exercises, prove or disprove the statements.

$\frac{\tan (x+y)}{1+\tan \,x\,\tan \,x}=\frac{\tan \,x+\tan \,y}{1-{\tan }^{2}x\,{\tan }^{2}y}$

For the following exercises, prove or disprove the statements.

If $\alpha ,\beta ,$ and $\gamma$ are angles in the same triangle, then prove or disprove $\sin (\alpha +\beta )=\sin \,\gamma .$

For the following exercises, prove or disprove the statements.

If $\alpha ,\beta ,$ and $y$ are angles in the same triangle, then prove or disprove $\tan \,\alpha +\tan \,\beta +\tan \,\gamma =\tan \,\alpha \,\tan \,\beta \,\tan \,\gamma$

Explain how to determine the reduction identities from the double-angle identity $\cos (2x)={\cos }^{2}x-{\sin }^{2}x.$