Problem 60 - Precalculus Essentials

Question

Use a sketch to find the exact value of each expression. $$\sec \left[\sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right)\right]$$

Step-by-Step Solution

Answer

The exact value of the expression \(\sec \left[\sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right)\right]\) is \(-\sqrt{2}\)

Step 1: Understand the Inverse Sine Function

The inverse sine function, denoted as \(\sin^{-1}(x)\), or arcsin(x), is the inverse of the sine function. \(\sin^{-1}\left(-\frac{\sqrt{2}}{2}\right)\) means the angle whose sine is \(-\frac{\sqrt{2}}{2}\). In a unit circle, such an angle is \(-\frac{\pi}{4}\)

Step 2: Calculating the Secant

The secant of an angle in a unit circle is the reciprocal of the cosine of the angle. After finding the angle from Step 1 to be \(-\frac{\pi}{4}\), we need to calculate the secant of this angle. The cosine of this angle is \(\sqrt{2}/2\), so its secant, being the reciprocal of the cosine, is \(\frac{2}{\sqrt{2}}\), which simplifies to \(\sqrt{2}\).

Step 3: Take Note of the Signs

While the magnitude of the secant of \(-\frac{\pi}{4}\) is \(\sqrt{2}\), it's important to also be aware of the sign. The secant is positive in the fourth quadrant (where our angle is), so the secant of \(-\frac{\pi}{4}\) is \(-\sqrt{2}\)

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry and serves as the backbone for understanding various trigonometric functions. Imagine a circle with a radius of exactly 1 unit centered at the origin of a coordinate plane. Each point on the circle's circumference corresponds to a coordinate \( (x, y) \) that represents the cosine and sine of an angle, respectively, that originates from the positive x-axis.

When we deal with the inverse sine function, or \( \sin^{-1} \) for short, we look for the angle whose sine value corresponds to a given number. This process essentially 'rewinds' from the sine value back to the angle. Using the unit circle, one can sketch and find the precise angle that correlates with a specific sine value. For example, an angle with the sine of \( -\frac{\sqrt{2}}{2} \) places it in the third or fourth quadrant, since sine values are negative there. Due to the symmetry of the circle, the related angle could be either \( \frac{3\pi}{4} \) or \( \frac{7\pi}{4} \) (also represented as \( -\frac{\pi}{4} \) since \( \sin(\frac{3\pi}{4}) = \sin(\frac{7\pi}{4}) \)).

Secant Function

The secant function, often abbreviated as 'sec', plays a lesser-known but still pivotal role in trigonometry. Secant is defined as the reciprocal of the cosine function, which means \( \sec(\theta) = \frac{1}{\cos(\theta)} \). But what does this mean on the unit circle? Instead of measuring the distance along the x-axis like cosine, secant measures the length of the line segment from the origin to a point on the circle's circumference that passes through \( (x, 1) \) or \( (x, -1) \) on the vertical line tangent to the circle.

When we calculate the secant of the angle found by using the inverse sine function, we are effectively looking for the length of this line segment. In the example from the exercise, after determining that the angle corresponding to \( \sin^{-1}\left(-\frac{\sqrt{2}}{2}\right) \) is \( -\frac{\pi}{4} \) or \( \frac{7\pi}{4} \) using the unit circle, we use the definition of secant to find that \( \sec(\theta) = \frac{2}{\sqrt{2}} \), which simplifies to \( \sqrt{2} \). As the fourth quadrant where \( -\frac{\pi}{4} \) lies has a positive x-coordinate, the secant (reciprocal of cosine) remains positive as well.

Sine and Cosine Relationships

The sine and cosine functions describe relationships between the sides of a right triangle and the angles within it, but they also reveal patterns within the unit circle. Understanding these relationships is critical when exploring trigonometric functions and their inverses.

Sine (\( \sin(\theta) \) ) corresponds to the y-coordinate on the unit circle for a given angle \( \theta \) and represents the ratio of the opposite side to the hypotenuse in a right triangle. Cosine (\( \cos(\theta) \) ), on the other hand, corresponds to the x-coordinate and represents the ratio of the adjacent side to the hypotenuse.

These two functions are fundamentally linked. In fact, they are co-functions, meaning \( \sin(\frac{\pi}{2} - \theta) = \cos(\theta) \), and vice versa. This relationship is evident on the unit circle, where they overlap and reflect symmetries between angles and their trigonometric values. The role of the inverse sine function is to reverse the original sine function, allowing us to determine angles when given the sine value. This interplay between sine, cosine, and their inverses is invaluable for solving a variety of trigonometric problems, like the one presented in our exercise, where understanding these relations assists in finding the secant of an angle after determining it through the inverse sine function.