Problem 55 - Precalculus Essentials

Question

Use a sketch to find the exact value of each expression. $$\sec \left[\sin ^{-1}\left(-\frac{1}{4}\right)\right]$$

Step-by-Step Solution

Answer

The exact value of the expression \(\sec \left[\sin ^{-1} \left(-\frac{1}{4}\right)\right]\) is \(\frac{4\sqrt{15}}{15}\).

Step 1: Calculate \(\sin^{-1}\)

The inverse sine of -1/4, denoted as \(\sin^{-1}(-1/4)\), is the angle whose sine is -1/4. Let this angle be \(\theta\). Therefore, \(\sin(\theta) = -1/4\). In a right-angled triangle, this means that the opposite side is -1 and the hypotenuse is 4.

Step 2: Determine The Adjacent Side Of The Angle

By Pythagorean theorem \(a^2 + b^2 = c^2\), where c is the hypotenuse and a, b are the other two sides. So \(b^2 = c^2 - a^2 = 4^2 - (-1)^2 = 15\). Thus, the length of the adjacent side (b) is \(\sqrt{15}\). Note that since \(\sin(\theta)\) is negative, \(\theta\) is in the fourth quadrant where the cosine value is positive.

Step 3: Calculate The Secant Of The Angle

Secant, \(\sec\), is the reciprocal of the cosine. So, \(\sec(\theta) = \frac{1}{\cos(\theta)}\). On a right-angled triangle, \(\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{\sqrt{15}}{4}\). Therefore the secant of the angle is the reciprocal of the cosine value, so \(\sec(\theta) = \frac{4}{\sqrt{15}} = \frac{4\sqrt{15}}{15}\)

Key Concepts

Secant

The secant function, or \( \sec(\theta) \), is an important trigonometric function used predominantly in right-angled triangles. Secant is essentially the reciprocal of the cosine function.

• That means \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
• When considering a right-angled triangle, the cosine of an angle \( \theta \) is defined as the ratio of the length of the adjacent side to the hypotenuse.
• Consequently, the secant is the ratio of the hypotenuse to the adjacent side.

In exercises like the one provided, where the inverse functions or arcsine values are present, understanding the relationship between secant and cosine properties becomes crucial. It helps to convert between these trigonometric expressions with ease and assists in solving for exact values through reciprocal functions. Therefore, knowing how to calculate the cosine can directly aid in determining the secant.

Pythagorean theorem

The Pythagorean theorem is a fundamental concept in geometry, especially when working with right-angled triangles. This theorem establishes a relationship between the lengths of the sides of a right-angled triangle.

• The formula is expressed as \( a^2 + b^2 = c^2 \), where \( c \) represents the hypotenuse (the longest side), and \( a \) and \( b \) are the other two sides.
• This allows you to find the length of any side, provided the lengths of the other two sides are known.

In our exercise, after determining the angle \( \theta \) using the inverse sine function, the Pythagorean theorem is applied to find the length of the adjacent side. In the triangle illustrated,

• The opposite side is \(-1\), and the hypotenuse is \(4\).
• By rearranging the theorem to solve for the adjacent side \( b \), we derive \( b^2 = 4^2 - (-1)^2 \), leading to \( b = \sqrt{15} \).

This application is highly useful while dealing with trigonometric identities and solving triangle-based problems.

Right-angled triangle

Right-angled triangles are a foundational concept in trigonometry, crucial for understanding trigonometric functions and identities. They are defined by one 90-degree angle, which introduces the fundamental concepts of sine, cosine, and secant.

• Each angle in a right triangle is related to the sides of the triangle through these functions.
• The hypotenuse is always opposite the right angle and is the longest side of the triangle.
• The opposite side is the side opposite the angle of interest, whereas the adjacent side is next to the angle and also adjacent to the right angle.

Thus, understanding right-angled triangles helps us relate the trigonometric functions to actual measurable lengths of triangles. This exercise emphasized finding the secant of an angle, which directly relates to these triangles:

• By determining where our angle lies in the coordinate plane (e.g., fourth quadrant),
• we found that certain trigonometric functions like cosine and thus secant would retain a specific sign.

This knowledge is applied when set against the backdrop of the Pythagorean theorem, enabling the full computation of trigonometric functions.