Section4.4Finding Angles

In our earlier work in Section 4.1 and Section 4.2, we observed that in any right triangle, if we know the measure of one additional angle and the length of one additional side, we can determine all of the other parts of the triangle. With the inverse trigonometric functions that we developed in Section 4.3, we are now also able to determine the missing angles in any right triangle where we know the lengths of two sides.

While the original trigonometric functions take a particular angle as input and provide an output that can be viewed as the ratio of two sides of a right triangle, the inverse trigonometric functions take an input that can be viewed as a ratio of two sides of a right triangle and produce the corresponding angle as output. Indeed, it's imperative to remember that statements such as

\begin{equation*} \arccos(x) = \theta \ \text{ and } \ \cos(\theta) = x \end{equation*}

say the exact same thing from two different perspectives, and that we read “$\arccos(x)$” as “the angle whose cosine is $x$”.

Preview Activity4.4.1.

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

1. Sketch a labeled picture of the triangle.

2. What is the exact length of the triangle's hypotenuse?

3. What is the exact value of $\sin(\theta)\text{?}$

4. Rewrite your equation from (b) using the arcsine function in the form $\arcsin(\Box) = \Delta\text{,}$ where $\Box$ and $\Delta$ are numerical values.

5. What special angle from the unit circle is $\theta\text{?}$

Subsection4.4.1Evaluating inverse trigonometric functions

Like the trigonometric functions themselves, there are a handful of important values of the inverse trigonometric functions that we can determine exactly without the aid of a computer. For instance, we know from the unit circle (Figure 2.3.1) that $\arcsin(-\frac{\sqrt{3}}{2}) = -\frac{\pi}{3}\text{,}$ $\arccos(-\frac{\sqrt{3}}{2}) = \frac{5\pi}{6}\text{,}$ and $\arctan(-\frac{1}{\sqrt{3}}) = -\frac{\pi}{6}\text{.}$ In these evaluations, we have to be careful to remember that the range of the arccosine function is $[0,\pi]\text{,}$ while the range of the arcsine function is $[-\frac{\pi}{2},\frac{\pi}{2}]$ and the range of the arctangent function is $(-\frac{\pi}{2},\frac{\pi}{2})\text{,}$ in order to ensure that we choose the appropriate angle that results from the inverse trigonometric function.

In addition, there are many other values at which we may wish to know the angle that results from an inverse trigonometric function. To determine such values, we use a computational device (such as Desmos) in order to evaluate the function.

Example4.4.1.

Consider the right triangle pictured in Figure 4.4.2 and assume we know that the vertical leg has length $1$ and the hypotenuse has length $3\text{.}$ Let $\alpha$ be the angle opposite the known leg. Determine exact and approximate values for all of the remaining parts of the triangle.

Solution

Because we know the hypotenuse and the side opposite $\alpha\text{,}$ we observe that $\sin(\alpha) = \frac{1}{3}\text{.}$ Rewriting this statement using inverse function notation, we have equivalently that $\alpha = \arcsin(\frac{1}{3})\text{,}$ which is the exact value of $\alpha\text{.}$ Since this is not one of the known special angles on the unit circle, we can find a numerical estimate of $\alpha$ using a computational device. Entering arcsin(1/3) in Desmos, we find that $\alpha \approx 0.3398$ radians. Note well: whatever device we use, we need to be careful to use degree or radian mode as dictated by the problem we are solving. We will always work in radians unless stated otherwise.

We can now find the remaining leg's length and the remaining angle's measure. If we let $x$ represent the length of the horizontal leg, by the Pythagorean Theorem we know that

\begin{equation*} x^2 + 1^2 = 3^2\text{,} \end{equation*}

and thus $x^2 = 8$ so $x = \sqrt{8} \approx 2.8284\text{.}$ Calling the remaining angle $\beta\text{,}$ since $\alpha + \beta = \frac{\pi}{2}\text{,}$ it follows that

\begin{equation*} \beta = \frac{\pi}{2} - \arcsin \left(\frac{1}{3}\right) \approx 1.2310\text{.} \end{equation*}
Activity4.4.2.

For each of the following different scenarios, draw a picture of the situation and use inverse trigonometric functions appropriately to determine the missing information both exactly and approximately.

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

2. 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 $\sin(\alpha)\text{?}$

3. 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 $\cos(\beta)\text{?}$

Subsection4.4.2Finding angles in applied contexts

Now that we have developed the (restricted) sine, cosine, and tangent functions and their respective inverses, in any setting in which we have a right triangle together with one side length and any one additional piece of information (another side length or a non-right angle measurement), we can determine all of the remaining pieces of the triangle. In the activities that follow, we explore these possibilities in a variety of different applied contexts.

Activity4.4.3.

A roof is being built with a “7-12 pitch.” This means that the roof rises $7$ inches vertically for every $12$ inches of horizontal span; in other words, the slope of the roof is $\frac{7}{12}\text{.}$ What is the exact measure (in degrees) of the angle the roof makes with the horizontal? What is the approximate measure? What are the exact and approximate measures of the angle at the peak of the roof (made by the front and back portions of the roof that meet to form the ridge)?

Activity4.4.4.

On a baseball diamond (which is a square with $90$-foot sides), the third baseman fields the ball right on the line from third base to home plate and $10$ feet away from third base (towards home plate). When he throws the ball to first base, what angle (in degrees) does the line the ball travels make with the first base line? What angle does it make with the third base line? Draw a well-labeled diagram to support your thinking.

What angles arise if he throws the ball to second base instead?

Activity4.4.5.

A camera is tracking the launch of a SpaceX rocket. The camera is located $4000$' from the rocket's launching pad, and the camera angle changes in order to keep the rocket in focus. At what angle $\theta$ (in radians) is the camera tilted when the rocket is $3000$' off the ground? Answer both exactly and approximately.

Now, rather than considering the rocket at a fixed height of $3000$', let its height vary and call the rocket's height $h\text{.}$ Determine the camera's angle, $\theta$ as a function of $h\text{,}$ and compute the average rate of change of $\theta$ on the intervals $[3000,3500]\text{,}$ $[5000,5500]\text{,}$ and $[7000,7500]\text{.}$ What do you observe about how the camera angle is changing?

Subsection4.4.3Summary

• Anytime we know two side lengths in a right triangle, we can use one of the inverse trigonometric functions to determine the measure of one of the non-right angles. For instance, if we know the values of $\text{opp}$ and $\text{adj}$ in Figure 4.4.3, then since

it follows that $\theta = \arctan(\frac{\text{opp}}{\text{adj}})\text{.}$

If we instead know the hypotenuse and one of the two legs, we can use either the arcsine or arccosine function accordingly.

• For situations other than angles or ratios that involve the $16$ special points on the unit circle, technology is required in order to evaluate inverse trignometric functions. For instance, from the unit circle we know that $\arccos(\frac{1}{2}) = \frac{\pi}{3}$ (exactly), but if we want to know $\arccos(\frac{1}{3})\text{,}$ we have to estimate this value using a computational device such as Desmos. We note that “$\arccos(\frac{1}{3})$” is the exact value of the angle whose cosine is $\frac{1}{3}\text{.}$

Exercises4.4.4Exercises

4.

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.

1. 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.)

2. What is the angle of depression when the plane is $6000$ feet from the building?

3. How far did the plane travel during the time between the two different observations?

4. What is the plane's velocity (in miles per hour)?

5.

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.

1. When the balloon is $200$ feet off the ground, what is the angle of elevation of the camera?

2. When the balloon is $275$ feet off the ground, what is the angle of elevation of the camera?

3. 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 $h\text{.}$

4. 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.

6.

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.

1. What is the exact value of $\cos(\alpha)\text{?}$

2. What is the exact value of $\sin(\beta)\text{?}$

3. What is the exact value of $\tan(\beta)\text{?}$ of $\tan(\alpha)\text{?}$

4. What is the exact radian measure of $\alpha\text{?}$ approximate measure?

5. What is the exact radian measure of $\beta\text{?}$ approximate measure?

6. True or false: for any two angles $\theta$ and $\gamma$ such that $\theta + \gamma = \frac{\pi}{2}$ (radians), it follows that $\cos(\theta) = \sin(\gamma)\text{.}$