Problem 23 - Precalculus with Calculus Previews

Question

Use the given information to find the values of the remaining five trigonometric functions. $$ \sin x=\frac{1}{3}, \quad \pi / 2

Step-by-Step Solution

Answer

\( \cos x = -\frac{2\sqrt{2}}{3} \), \( \tan x = -\frac{\sqrt{2}}{4} \), \( \csc x = 3 \), \( \sec x = -\frac{3\sqrt{2}}{4} \), \( \cot x = -2\sqrt{2} \).

Step 1: Understanding the Quadrant

Since the angle \(x\) is in the range \(\frac{\pi}{2} < x < \pi\), it lies in the second quadrant. In the second quadrant, the sine function is positive while cosine and tangent functions are negative.

Step 2: Find Cosine Value

To find the cosine value, we use the Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \). Substituting \( \sin x = \frac{1}{3} \), we have:\[ \left(\frac{1}{3}\right)^2 + \cos^2 x = 1 \]\[ \frac{1}{9} + \cos^2 x = 1 \]\[ \cos^2 x = 1 - \frac{1}{9} = \frac{8}{9} \]Taking the square root, we find \( \cos x = -\sqrt{\frac{8}{9}} = -\frac{2\sqrt{2}}{3} \), because cosine is negative in the second quadrant.

Step 3: Calculate Tangent Value

Tangent is the ratio of sine to cosine. Thus,\[ \tan x = \frac{\sin x}{\cos x} = \frac{\frac{1}{3}}{-\frac{2\sqrt{2}}{3}} = -\frac{1}{2\sqrt{2}} \]Simplifying gives \( \tan x = -\frac{\sqrt{2}}{4} \).

Step 4: Determine Cosecant Value

Cosecant is the reciprocal of sine. Therefore, \( \csc x = \frac{1}{\sin x} = \frac{1}{\frac{1}{3}} = 3 \).

Step 5: Determine Secant Value

Secant is the reciprocal of cosine. Thus, \( \sec x = \frac{1}{\cos x} = \frac{1}{-\frac{2\sqrt{2}}{3}} = -\frac{3}{2\sqrt{2}} \). Simplifying gives \( \sec x = -\frac{3\sqrt{2}}{4} \).

Step 6: Determine Cotangent Value

Cotangent is the reciprocal of tangent. So, \( \cot x = \frac{1}{\tan x} = \frac{1}{-\frac{\sqrt{2}}{4}} = -\frac{4}{\sqrt{2}} \). Simplifying gives \( \cot x = -2\sqrt{2} \).

Key Concepts

Sine Function

The sine function, often represented as \( \sin x \), is one of the fundamental trigonometric functions. It is vital in understanding angles and right triangles. The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. Mathematically, it's expressed as:\[ \sin x = \frac{\text{Opposite Side}}{\text{Hypotenuse}}\]This definition helps us solve many problems involving distances and angles. In the unit circle, the sine value at any angle \( x \) is represented by the y-coordinate of the corresponding point on the circle.

Let's consider the exercise in which \( \sin x = \frac{1}{3} \) in the second quadrant. This specifies that regardless of the actual triangle size, the ratio will remain constant. It is essential to know that in the second quadrant, the sine function remains positive, aligning with our initial problem statement.

Cosine Function

The cosine function, symbolized by \( \cos x \), is another key trigonometric function. It is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle:\[ \cos x = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}\]In the context of the unit circle, \( \cos x \) corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle.

In the given problem, utilizing the Pythagorean identity helped find \( \cos x \) using the equation \( \sin^2 x + \cos^2 x = 1 \). With \( \sin x = \frac{1}{3} \), it becomes:\[ \left(\frac{1}{3}\right)^2 + \cos^2 x = 1\]Solving, we get \( \cos x = -\frac{2\sqrt{2}}{3} \). It's negative because cosine is negative in the second quadrant.

Pythagorean Identity

The Pythagorean identity is a crucial tool in trigonometry that relates the square of sine and cosine of an angle.
This identity is expressed as:\[ \sin^2 x + \cos^2 x = 1\]This equation stems from the Pythagorean theorem and holds true for any angle in the unit circle, making it widely used to find one trigonometric ratio if the other is known.

In our step-by-step solution, we effectively used this identity to determine \( \cos x \) when \( \sin x \) was provided. By rearranging the formula, we deduced the expression:\[ \cos^2 x = 1 - \sin^2 x\]Substituting the values, we confirmed \( \cos x = -\frac{2\sqrt{2}}{3} \). The identity is pivotal in ensuring our calculations align with the geometric concepts.