삼각함수 전체 문제

For the following exercises, prove the identity.

$\frac{\cos (3x)+\cos \,x}{\cos (3x)-\cos \,x}=-\cot \,(2x)\cot \,x$

For the following exercises, prove the identity.

$\frac{\cos (6y)+\cos (8y)}{\sin (6y)-\sin (4y)}=\cot \,y\,\cos \,(7y)\sec \,(5y)$

For the following exercises, prove the identity.

$\frac{\cos (2y)-\cos (4y)}{\sin (2y)+\sin (4y)}=\tan \,y$

For the following exercises, prove the identity.

$\frac{\sin (10x)-\sin (2x)}{\cos (10x)+\cos (2x)}=\tan (4x)$

For the following exercises, prove the identity.

$\cos \,x-\cos (3x)=4\,{\sin }^{2}x\cos \,x$

For the following exercises, prove the identity.

${(\cos (2x)-\cos (4x))}^{2}+{(\sin (4x)+\sin (2x))}^{2}=4\,{\sin }^{2}(3x)$

For the following exercises, prove the identity.

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

For the following exercises, prove the identity.

$\cos (4x)-\cos (3x)\cos x={\sin }^{2}x-4\,{\cos }^{2}x\,{\sin }^{2}x$

For the following exercises, prove the identity.

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

For the following exercises, prove the identity.

$\frac{2\cos (2x)}{\sin (2x)}=\cot \,x-\tan \,x$

For the following exercises, prove the identity.

$\cot \,x\,\cos (2x)=-\sin (2x)+\cot \,x$

For the following exercises, prove the identity.

${\tan }^{3}x-\tan \,x\,{\sec }^{2}x=\tan (-x)$

For the following exercises, prove the identity.

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

For the following exercises, prove the identity.

$\frac{\sin (2x)}{\sin \,x}-\frac{\cos (2x)}{\cos \,x}=\sec \,x$

For the following exercises, prove the identity.

Plot the points and find a function of the form $y=A\cos (Bx+C)+D$ that fits the given data.

For the following exercises, prove the identity.

The displacement $h(t)$ in centimeters of a mass suspended by a spring is modeled by the function $h(t)=\frac{1}{4}\,\sin (120\pi t),$ where $t$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

For the following exercises, prove the identity.

A woman is standing 300 feet away from a 2000-foot building. If she looks to the top of the building, at what angle above horizontal is she looking? A worker looks down at her from the 15^th floor (1500 feet above her). At what angle is he looking down at her? Round to the nearest tenth of a degree.

For the following exercises, prove the identity.

Two frequencies of sound are played on an instrument governed by the equation $n(t)=8\,\cos (20\pi t)\cos (1000\pi t).$ What are the period and frequency of the “fast” and “slow” oscillations? What is the amplitude?

For the following exercises, prove the identity.

The average monthly snowfall in a small village in the Himalayas is 6 inches, with the low of 1 inch occurring in July. Construct a function that models this behavior. During what period is there more than 10 inches of snowfall?

For the following exercises, prove the identity.

A spring attached to a ceiling is pulled down 20 cm. After 3 seconds, wherein it completes 6 full periods, the amplitude is only 15 cm. Find the function modeling the position of the spring $t$ seconds after being released. At what time will the spring come to rest? In this case, use 1 cm amplitude as rest.