삼각함수 전체 문제

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.

For the following exercises, prove the identity.

Water levels near a glacier currently average 9 feet, varying seasonally by 2 inches above and below the average and reaching their highest point in January. Due to global warming, the glacier has begun melting faster than normal. Every year, the water levels rise by a steady 3 inches. Find a function modeling the depth of the water $t$ months from now. If the docks are 2 feet above current water levels, at what point will the water first rise above the docks?

Describe the altitude of a triangle.

Compare right triangles and oblique triangles.

When can you use the Law of Sines to find a missing angle?

In the Law of Sines, what is the relationship between the angle in the numerator and the side in the denominator?

What type of triangle results in an ambiguous case?

For the following exercises, assume $\alpha$ is opposite side $a,\beta$ is opposite side $b,$ and $\gamma$ is opposite side $c.$ Solve each triangle, if possible. Round each answer to the nearest tenth.

$\alpha =43^\circ ,\gamma =69^\circ ,a=20$

For the following exercises, assume $\alpha$ is opposite side $a,\beta$ is opposite side $b,$ and $\gamma$ is opposite side $c.$ Solve each triangle, if possible. Round each answer to the nearest tenth.

$\alpha =35^\circ ,\gamma =73^\circ ,c=20$

For the following exercises, assume $\alpha$ is opposite side $a,\beta$ is opposite side $b,$ and $\gamma$ is opposite side $c.$ Solve each triangle, if possible. Round each answer to the nearest tenth.

$\alpha =60^\circ ,$ $\beta =60^\circ ,$ $\gamma =60^\circ$

For the following exercises, assume $\alpha$ is opposite side $a,\beta$ is opposite side $b,$ and $\gamma$ is opposite side $c.$ Solve each triangle, if possible. Round each answer to the nearest tenth.

$a=4,$ $\alpha =\,60^\circ ,$ $\beta =100^\circ$

For the following exercises, assume $\alpha$ is opposite side $a,\beta$ is opposite side $b,$ and $\gamma$ is opposite side $c.$ Solve each triangle, if possible. Round each answer to the nearest tenth.

$b=10,$ $\beta =95^\circ ,\gamma =\,30^\circ$

For the following exercises, use the Law of Sines to solve for the missing side for each oblique triangle. Round each answer to the nearest hundredth. Assume that angle $A$ is opposite side $a,$ angle $B$ is opposite side $b,$ and angle $C$ is opposite side $c.$

Find side $b$ when $A=37^\circ ,$ $B=49^\circ ,$ $c=5.$

For the following exercises, use the Law of Sines to solve for the missing side for each oblique triangle. Round each answer to the nearest hundredth. Assume that angle $A$ is opposite side $a,$ angle $B$ is opposite side $b,$ and angle $C$ is opposite side $c.$

Find side $a$ when $A=132^\circ ,C=23^\circ ,b=10.$

For the following exercises, use the Law of Sines to solve for the missing side for each oblique triangle. Round each answer to the nearest hundredth. Assume that angle $A$ is opposite side $a,$ angle $B$ is opposite side $b,$ and angle $C$ is opposite side $c.$

Find side $c$ when $B=37^\circ ,C=21^\circ ,$ $b=23.$