Suppose that a star forms an angle $\theta$ with respect to Earth and the sun. Let the coordinates of Earth be $(x, y)$, those of the star $(0,0)$, and those of the sun $(x, 0)$. Find an equation for $x$, the distance between the sun and the star, as follows.

Solve your equation for $x$.

To correct mild double vision, a small amount of prism is added to a patient's eyeglasses. The amount of light shift this causes is measured in prism diopters. A patient needs $12$ prism diopters horizontally and $5$ prism diopters vertically. A prism that corrects for both requirements should have length $r$ and be set at angle $\theta$.

Use the Pythagorean theorem to find $r$.

Find all values of $\theta$, if $\theta$ is in the interval $\left[0^{\circ}, 360^{\circ}\right)$ and has the given function value. See the example in this section.

$$\cot \theta=-\dfrac{\sqrt{3}}{3}$$

Find all values of $\theta$, if $\theta$ is in the interval $\left[0^{\circ}, 360^{\circ}\right)$ and has the given function value. See the example in this section.

$$\sec \theta=-\sqrt{2}$$

Use a trigonometric function ratio to solve each problem.

A civil engineer must determine the height of the tree. The given angle was measured with a clinometer. She knows that $\sin 70^{\circ} \approx .9397, \cos 70^{\circ} \approx .3420$, and $\tan 70^{\circ} \approx 2.747$. Use the pertinent trigonometric function and the measurement to find the height of the tree to the nearest whole number.

The base is $50 \mathrm{~ft}$.

Find all values of $\theta$, if $\theta$ is in the interval $\left[0^{\circ}, 360^{\circ}\right)$ and has the given function value. See the example in this section.

$$\tan \theta=-\sqrt{3}$$

True or false: Since $\cot \theta=\dfrac{\cos \theta}{\sin \theta}$, if $\cot \theta=\dfrac{1}{2}$ with $\theta$ in quadrant $\mathrm{I}$, then $\cos \theta=1$ and $\sin \theta=2$. If false, explain why.

True or false: For all angles $\theta, \sin \theta+\cos \theta=1$. If false, give an example showing why it is false.

Find all values of $\theta$, if $\theta$ is in the interval $\left[0^{\circ}, 360^{\circ}\right)$ and has the given function value. See the example in this section.

$$\cos \theta=\dfrac{\sqrt{3}}{2}$$

Find all values of $\theta$, if $\theta$ is in the interval $\left[0^{\circ}, 360^{\circ}\right)$ and has the given function value. See the example in this section.

$$\sin \theta=\dfrac{1}{2}$$

Work each problem.

Using a method similar to the one given in this section showing that $\dfrac{\sin \theta}{\cos \theta}=\tan \theta$, show that $\dfrac{\cos \theta}{\sin \theta}=\cot \theta$.

Find a values of $\theta$ in $\left[0^{\circ}, 90^{\circ}\right]$ that satisfies each statement. Leave answers in decimal degrees. See the example in this section.

$$\cot \theta=1.2575516$$

Work each problem.

Derive the identity $1+\cot ^2 \theta=\csc ^2 \theta$ by dividing $x^2+y^2=r^2$ by $y^2$.

Find all trigonometric function values for each angle. Use a calculator. See the examples in this section.

$\sin \theta=.164215$, with $\theta$ in quadrant $\mathrm{II}$