Problem 17 - Precalculus: Functions and Graphs

Question

Find the exact values of the trigonometric functions for the acute angle \(\theta\). $$\sin \theta=\frac{3}{5}$$

Step-by-Step Solution

Answer

The values are \( \cos \theta = \frac{4}{5} \) and \( \tan \theta = \frac{3}{4} \).

Step 1: Understand the Problem

We are given \( \sin \theta = \frac{3}{5} \) and need to find the exact values of \( \cos \theta \) and \( \tan \theta \) for an acute angle \( \theta \).

Step 2: Use the Pythagorean Identity

Recall the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \sin \theta = \frac{3}{5} \) into this identity: \( \left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1 \).

Step 3: Simplify the Equation

Calculate \( \sin^2 \theta = \frac{9}{25} \). Then, substitute it into the identity: \( \frac{9}{25} + \cos^2 \theta = 1 \).

Step 4: Isolate \( \cos^2 \theta \)

Subtract \( \frac{9}{25} \) from both sides: \( \cos^2 \theta = 1 - \frac{9}{25} \). Simplify the right side: \( \cos^2 \theta = \frac{16}{25} \).

Step 5: Solve for \( \cos \theta \)

Take the square root of both sides. Since \( \theta \) is an acute angle, \( \cos \theta \) is positive: \( \cos \theta = \frac{4}{5} \).

Step 6: Calculate \( \tan \theta \)

Use the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substitute the known values: \( \tan \theta = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \).

Step 7: Summarize the Trigonometric Values

We have found \( \cos \theta = \frac{4}{5} \) and \( \tan \theta = \frac{3}{4} \) for the angle \( \theta \).

Key Concepts

Pythagorean Identity

In trigonometry, the Pythagorean identity is a fundamental concept that connects the squares of sine and cosine functions. This identity is expressed as \( \sin^2 \theta + \cos^2 \theta = 1 \). It's derived from the Pythagorean theorem because the sine and cosine values of an angle in a right triangle can be thought of as the lengths of the sides of the triangle, with the hypotenuse being 1.
When you know the sine of an angle, as in the exercise where \( \sin \theta = \frac{3}{5} \), you can use this identity to find the cosine of the angle. Simply substitute \( \sin \theta \) into the identity, square it, and solve for \( \cos^2 \theta \).
This step is crucial as it helps in determining the cosine value that satisfies both the trigonometric setup and the condition that the given angle is acute.
To summarize, the Pythagorean identity allows you to find relationships between sine and cosine, and solve for one when you know the other.

Acute Angle

An acute angle is any angle less than 90 degrees. In the context of trigonometry, knowing that an angle is acute helps determine the signs of the trigonometric functions associated with it.
Sine and cosine values for acute angles are always positive, as they are depicted on the unit circle in the first quadrant, where all angle measures are between 0 and 90 degrees.
In the problem at hand, we have been given \( \sin \theta = \frac{3}{5} \), with the information that \( \theta \) is acute. This allows us to posit that both \( \cos \theta \) and other related functions, like \( \tan \theta \), will also yield positive results.
Understanding this concept is vital when working with trigonometric functions since the positivity or negativity of these functions can be influenced by the angle's quadrant.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables where both sides of the equation are defined. They serve as useful tools for simplifying expressions and solving trigonometric equations.
The primary identities you might encounter include:

• The Pythagorean identities: \( \sin^2 \theta + \cos^2 \theta = 1 \), \( 1+ \tan^2 \theta = \sec^2 \theta \), and \( 1 + \cot^2 \theta = \csc^2 \theta \).
• The sum and difference formulas, like \( \sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \).
• Double-angle formulas such as \( \sin(2\theta) = 2 \sin \theta \cos \theta \).

In the original exercise, we specifically use the Pythagorean identity and the definition of tangent based on sine and cosine. \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) is used to find the tangent of an angle when sine and cosine are already known.
These identities not only expand your toolkit for solving complex trigonometric problems but also improve your understanding of the relationships between the different trigonometric functions.