4.4 Finding Angles

Introduction

Evaluating inverse trigonometric functions

Finding angles in applied contexts

Summary

Consider a right triangle that has one leg of length $3$ and another leg of length $\sqrt{3}$. Let $\theta$ be the angle that lies opposite the shorter leg.

Sketch a labeled picture of the triangle.

Consider the right triangle pictured in Figure and assume we know that the vertical leg has length $1$ and the hypotenuse has length $3$. Let $\alpha$ be the angle opposite the known leg. Determine exact and approximate values for all of the remaining parts of the triangle.

Consider a right triangle with legs of length $11$ and $13$. What are the measures (in radians) of the non-right angles and what is the length of the hypotenuse?

Consider an angle $\alpha$ in standard position (vertex at the origin, one side on the positive $x$-axis) for which we know $\cos(\alpha) = -\frac{1}{2}$ and $\alpha$ lies in quadrant III. What is the measure of $\alpha$ in radians? In addition, what is the value of $\sin(\alpha)$?

Consider an angle $\beta$ in standard position for which we know $\sin(\beta) = 0.1$ and $\beta$ lies in quadrant II. What is the measure of $\beta$ in radians? In addition, what is the value of $\cos(\beta)$?

At an airshow, a pilot is flying low over a runway while maintaining a constant altitude of $2000$ feet and a constant speed. On a straight path over the runway, the pilot observes on her laser range-finder that the distance from the plane to a fixed building adjacent to the runway is $7500$ feet. Five seconds later, she observes that distance to the same building is now $6000$ feet.

• What is the angle of depression from the plane to the building when the plane is $7500$ feet away from the building? (The angle of depression is the angle that the pilot's line of sight makes with the horizontal.) - What is the angle of depression when the plane is $6000$ feet from the building? - How far did the plane travel during the time between the two different observations? - What is the plane's velocity (in miles per hour)?

On a calm day, a photographer is filming a hot air balloon. When the balloon launches, the photographer is stationed $850$ feet away from the balloon.

• When the balloon is $200$ feet off the ground, what is the angle of elevation of the camera? - When the balloon is $275$ feet off the ground, what is the angle of elevation of the camera? - Let $\theta$ represent the camera's angle of elevation when the balloon is at an arbitrary height $h$ above the ground. Express $\theta$ as a function of $h$. - Determine $AV_{[200,275]}$ for $\theta$ (as a function of $h$) and write at least one sentence to carefully explain the meaning of the value you find, including units.

Consider a right triangle where the two legs measure $5$ and $12$ respectively and $\alpha$ is the angle opposite the shorter leg and $\beta$ is the angle opposite the longer leg.

• What is the exact value of $\cos(\alpha)$? - What is the exact value of $\sin(\beta)$? - What is the exact value of $\tan(\beta)$? of $\tan(\alpha)$? - What is the exact radian measure of $\alpha$? approximate measure? - What is the exact radian measure of $\beta$? approximate measure? - True or false: for any two angles $\theta$ and $\gamma$ such that $\theta + \gamma = \frac{\pi}{2}$ (radians), it follows that $\cos(\theta) = \sin(\gamma)$.