삼각함수의 뜻과 관계

For the following exercises, find the requested information.

Given that $\sin \,a=\frac{2}{3}$ and $\cos \,b=-\frac{1}{4},$ with $a$ and $b$ both in the interval $[\frac{\pi }{2},\pi ),$ find $\sin (a+b)$ and $\cos (a-b).$

For the following exercises, find the requested information.

Given that $\sin \,a=\frac{4}{5},$ and $\cos \,b=\frac{1}{3},$ with $a$ and $b$ both in the interval $[0,\frac{\pi }{2}),$ find $\sin (a-b)$ and $\cos (a+b).$

For the following exercises, find the exact value of each expression.

$\sin \left({\cos }^{-1}(0)-{\cos }^{-1}\left(\frac{1}{2}\right)\right)$

For the following exercises, find the exact value of each expression.

$\cos \left({\cos }^{-1}\left(\frac{\sqrt{2}}{2}\right)+{\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)\right)$

For the following exercises, find the exact value of each expression.

$\tan \left({\sin }^{-1}\left(\frac{1}{2}\right)-{\cos }^{-1}\left(\frac{1}{2}\right)\right)$

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.

$\tan (-{15}^{\circ })$

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

Explain how to determine the double-angle formula for $\tan (2x)$ using the double-angle formulas for $\cos (2x)$ and $\sin (2x).$

We can determine the half-angle formula for $\tan \left(\frac{x}{2}\right)=\frac{\sqrt{1-\cos \,x}}{\sqrt{1+\cos \,x}}$ by dividing the formula for $\sin \left(\frac{x}{2}\right)$ by $\cos \left(\frac{x}{2}\right).$ Explain how to determine two formulas for $\tan \left(\frac{x}{2}\right)$ that do not involve any square roots.

For the half-angle formula given in the previous exercise for $\tan \left(\frac{x}{2}\right),$ explain why dividing by 0 is not a concern. (Hint: examine the values of $\cos \,x$ necessary for the denominator to be 0.)

For the following exercises, find the exact values of a) $\sin (2x),$ b) $\cos (2x),$ and c) $\tan (2x)$ without solving for $x.$

If $\sin \,x=\frac{1}{8},$ and $x$ is in quadrant I.

For the following exercises, find the exact values of a) $\sin (2x),$ b) $\cos (2x),$ and c) $\tan (2x)$ without solving for $x.$

If $\cos \,x=\frac{2}{3},$ and $x$ is in quadrant I.

For the following exercises, find the exact values of a) $\sin (2x),$ b) $\cos (2x),$ and c) $\tan (2x)$ without solving for $x.$

If $\cos \,x=-\frac{1}{2},$ and $x$ is in quadrant III.

For the following exercises, find the exact values of a) $\sin (2x),$ b) $\cos (2x),$ and c) $\tan (2x)$ without solving for $x.$

If $\tan \,x=-8,$ and $x$ is in quadrant IV.

For the following exercises, find the values of the six trigonometric functions if the conditions provided hold.

$\cos (2\theta )=\frac{3}{5}$ and ${90}^{\circ }\le \theta \le {180}^{\circ }$

For the following exercises, find the values of the six trigonometric functions if the conditions provided hold.

$\cos (2\theta )=\frac{1}{\sqrt{2}}$ and ${180}^{\circ }\le \theta \le {270}^{\circ }$