Question

In Exercises \(27-30\) , give the measure of the angle in radians and degrees. Give exact answers whenever possible. $$\cos ^{-1}(0.7)$$

Short answer

After performing the calculations, you should find that \(\cos^{-1}(0.7)\) yields an angle measure of approximately \(0.795 rad\) or \(45.57^{\circ}\). These are both rounded to the nearest hundredth for simplicity.

Step-by-step solution

Use Inverse Cosine

First, find the measure of the angle in radians using the inverse cosine function on the provided value, which is 0.7. According to the concept of inverse trigonometric functions, \(\cos^{-1}(0.7)\) gives the angle so that \(\cos(\angle) = 0.7\). This can be calculated directly with a scientific calculator.

Convert Radians to Degrees

Next, convert the obtained radian measure to degrees. Recall that \(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\), and use this conversion factor to do the calculation. Multiply the angle in radians with \(\frac{180}{\pi}\) to get the result in degrees.

Key concepts

Inverse Cosine

When a student encounters the term inverse cosine, also denoted by \(\cos^{-1}\), it's essential to understand it as a function that performs the reverse operation of the cosine function. Whenever you're given a cosine value, the inverse cosine will tell you the angle from which it was derived.

For instance, applying \(\cos^{-1}(0.7)\) informs us about the angle whose cosine is 0.7. This is particularly useful in solving geometry and trigonometry problems, where we often know the lengths of sides in a right triangle and need to find the angle measures. In the provided exercise, when we calculate \(\cos^{-1}(0.7)\), a scientific calculator or a software tool will yield the angle in radians, which is the standard unit of angle in most mathematical settings.

Angle Measurement in Radians

Now let's delve into the concept of angle measurement in radians. Unlike degrees, which may be more familiar from everyday usage, radians provide a way to relate arc lengths to the radius of a circle in a more mathematically consistent manner. One radian is the angle at the center of a circle that spans an arc equal in length to the radius of the circle.

To visualize this, imagine wrapping the radius of a circle around its circumference. The angle subtended by that segment is one radian. There are \(2\pi\) radians in a full circle, equivalent to 360 degrees. This connection between the circumference of a circle and the radius is crucial in trigonometry and calculus, as it results in more natural integration of the trigonometric functions with other areas of mathematics.

Radian to Degree Conversion

Converting from radian to degree measurements and vice versa is a fundamental skill in trigonometry. Students often need to switch between these units to interpret angle measures in different contexts. The essential conversion factor to remember is \(1\) radian = \(\frac{180}{\pi}\) degrees.

To convert from radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\). Conversely, to convert from degrees to radians, you'll multiply the degree measure by \(\frac{\pi}{180}\). For the inverse cosine value \(\cos^{-1}(0.7)\), once we have the angle in radians, the multiplication by \(\frac{180}{\pi}\) will yield the angle in degrees. This practical tool allows students to not only solve problems with ease but also to deepen their understanding of the interconnected nature of angle measures.

Exact Values of Trigonometric Functions

Understanding exact values of trigonometric functions can be incredibly helpful for students. These are the trigonometric values that can be computed exactly, without approximating, and are commonly found for angles like 0, \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\) radians. For example, the cosine of \(\frac{\pi}{4}\) radians is \(\frac{\sqrt{2}}{2}\), which is an exact value.

When solving trigonometric equations involving inverse functions, if the result is one of these commonly known angles, the exact value will be more favorable than a decimal approximation. This understanding is not only essential for precise calculations but also for verifying solutions in various mathematical problems. Learning the exact values also aids in memorizing the unit circle, which is central to trigonometry.