Problem 66 - Precalculus: A Concise Course

Question

Find the exact value of the expression. (Hint: Sketch a right triangle.) $$ \sec \left[\sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right)\right] $$

Step-by-Step Solution

Answer

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

Step 1: Calculate the inverse sine

First, calculate \( \sin^{-1}\left(-\frac{\sqrt{2}}{2}\right) \). With reference to the unit circle, \( \sin^{-1}\left(-\frac{\sqrt{2}}{2}\right) \) corresponds to an angle of -45° or -π/4 in radians because the sine of an angle is negative in the third and fourth quadrants and it equals \(-\frac{\sqrt{2}}{2}\) at both these angles.

Step 2: Draw a right triangle

Then, draw a right triangle that represents this angle. The opposite side will be -√2 (because of the negative sine) and the hypotenuse will be 2.

Step 3: Apply the Pythagorean theorem

Next, apply the Pythagorean theorem to these sides to find the adjacent side. The theorem gives us that the square of the hypotenuse (2^2) minus the square of the opposite side (-√2^2) equals the square of the adjacent side, which gives us an adjacent side of √2.

Step 4: Calculate the secant

Finally, calculate the secant of the angle, which is the ratio of the hypotenuse to the adjacent side. This gives us \( \sec(-π/4) = 2/√2 = √2 \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are tools used to find angles when given the value of a trigonometric ratio. These functions are essential when you want to determine an angle with a specified trigonometric value. For instance, the inverse sine function, denoted as \( \sin^{-1} \) (or arcsin), tells us the angle whose sine is a given number. It's crucial to know that the range of \( \sin^{-1}(x) \) is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
Here, in our problem \( \sin^{-1} \left(-\frac{\sqrt{2}}{2}\right) \), we're looking for an angle where the sine is \(-\frac{\sqrt{2}}{2}\). This angle corresponds to \(-\frac{\pi}{4}\) radians, taking into account that sine is negative in the fourth quadrant. Understanding these concepts helps us decode angles from trigonometric expressions effectively.

Unit Circle

The unit circle is a fundamental concept that helps visualize trigonometric functions for different angles. It's a circle with a radius of 1, centered at the origin of a coordinate system. Angles are measured in radians and can be positive or negative, with full rotation equaling \(2\pi\).
In our example, knowing \( \sin^{-1} \left(-\frac{\sqrt{2}}{2}\right) \) equates to an angle of \(-\frac{\pi}{4}\), we consider the unit circle. Here, the angle \(-\frac{\pi}{4}\) is located in the fourth quadrant where sine values are negative, which matches our requirement.
Using the unit circle is advantageous as it provides a solid visual context for understanding the values of sine, cosine, and other trigonometric functions at various angles.

Right Triangle Geometry

Right triangle geometry is another key pillar in solving trigonometric problems. This involves understanding the relationships between the triangle's sides and its angles. In the context of trigonometry, the functions sine, cosine, and tangent are based on these side ratios.
In this exercise, imagining a right triangle helps to make sense of \( \sin^{-1} \left(-\frac{\sqrt{2}}{2}\right) \). A triangle where one angle is \(-\frac{\pi}{4}\), allows us to set the opposite side \(-\sqrt{2}\), and the hypotenuse as \(2\). Right triangle geometry guides us to draw these associations clearly, making complex-looking expressions much easier to interpret and solve.

Pythagorean Theorem

The Pythagorean Theorem is a cornerstone of geometry, especially useful in dealing with right triangles. It states that in a right-angled triangle, the square of the hypotenuse's length is equal to the sum of the squares of the other two sides. Mathematically, it's expressed as \(a^2 + b^2 = c^2\).
For our problem, the We used this theorem to find the missing side of the triangle with sides \(-\sqrt{2}\) (opposite) and \(2\) (hypotenuse). Rearranging the formula gives \[c^2 - (-\sqrt{2})^2 = a^2,\] simplifying to find the adjacent side as \(\sqrt{2}\). Thus, the Pythagorean Theorem is pivotal in determining the exact measures of triangle sides necessary to calculate root functions like the secant in our problem.