삼각함수의 뜻과 관계

For the following exercises, simplify the given expression.

$\sin (2x)\,\cos (5x)-\sin (5x)\,\cos (2x)$

For the following exercises, simplify the given expression.

$\frac{\tan \left(\frac{3}{2}x\right)-\tan \left(\frac{7}{5}x\right)}{1+\tan \left(\frac{3}{2}x\right)\tan \left(\frac{7}{5}x\right)}$

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, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. Confirm your answer using a graphing calculator.

$\cos \left(\frac{\pi }{2}-x\right)$

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. Confirm your answer using a graphing calculator.

$\sin (\pi -x)$

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. Confirm your answer using a graphing calculator.

$\tan \left(\frac{\pi }{3}+x\right)$

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. Confirm your answer using a graphing calculator.

$\sin \left(\frac{\pi }{3}+x\right)$

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. Confirm your answer using a graphing calculator.

$\tan \left(\frac{\pi }{4}-x\right)$

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. Confirm your answer using a graphing calculator.

$\cos \left(\frac{7\pi }{6}+x\right)$

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. Confirm your answer using a graphing calculator.

$\sin \left(\frac{\pi }{4}+x\right)$

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. Confirm your answer using a graphing calculator.

$\cos \left(\frac{5\pi }{4}+x\right)$

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 $2x=x+x.$ )

$f(x)=\sin (4x)-\sin (3x)\cos \,x,g(x)=\sin \,x\,\cos (3x)$

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 $2x=x+x.$ )

$f(x)=\cos (4x)+\sin \,x\,\sin (3x),g(x)=-\cos \,x\,\cos (3x)$

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 $2x=x+x.$ )

$f(x)=\sin (3x)\cos (6x),g(x)=-\sin (3x)\cos (6x)$

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 $2x=x+x.$ )

$f(x)=\sin (4x),g(x)=\sin (5x)\cos \,x-\cos (5x)\sin \,x$

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 $2x=x+x.$ )

$f(x)=\sin (2x),g(x)=2\,\sin \,x\,\cos \,x$