Problem 17 - Precalculus: Functions and Graphs

Question

If \(\sin \alpha=-\frac{5}{13}\) and \(\tan \alpha>0,\) find the exact value of \(\sin \left(\alpha-\frac{\pi}{3}\right)\)

Step-by-Step Solution

Answer

\( \sin \left(\alpha - \frac{\pi}{3}\right) = \frac{-5 + 12\sqrt{3}}{26} \)

Step 1: Identify the Quadrant

Since \( \sin \alpha = -\frac{5}{13} \) and \( \tan \alpha > 0 \), it indicates that \( \alpha \) is in the third quadrant where both sine is negative and tangent is positive.

Step 2: Find \( \cos \alpha \)

Using the identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \), substitute \( \sin \alpha = -\frac{5}{13} \):\[ (-\frac{5}{13})^2 + \cos^2 \alpha = 1 \]\[ \frac{25}{169} + \cos^2 \alpha = 1 \]\[ \cos^2 \alpha = 1 - \frac{25}{169} \]\[ \cos^2 \alpha = \frac{144}{169} \]Thus, \( \cos \alpha = -\frac{12}{13} \), as \( \alpha \) is in the third quadrant where cosine is negative.

Step 3: Use the Angle Subtraction Formula for Sine

The formula for sine difference is \( \sin(a - b) = \sin a \cos b - \cos a \sin b \).For \( \sin \left(\alpha - \frac{\pi}{3}\right) \), let \( a = \alpha \) and \( b = \frac{\pi}{3} \).Thus, \[ \sin \left(\alpha - \frac{\pi}{3}\right) = \sin \alpha \cos \frac{\pi}{3} - \cos \alpha \sin \frac{\pi}{3} \]

Step 4: Calculate \( \sin \left(\alpha - \frac{\pi}{3}\right) \)

Substitute the known values into the formula:\[ \sin \left(\alpha - \frac{\pi}{3}\right) = \left(-\frac{5}{13}\right) \cdot \frac{1}{2} - \left(-\frac{12}{13}\right) \cdot \frac{\sqrt{3}}{2} \]\[ = -\frac{5}{26} + \frac{12\sqrt{3}}{26} \]\[ = \frac{-5 + 12\sqrt{3}}{26} \]

Step 5: Simplify the Result

The expression \( \frac{-5 + 12\sqrt{3}}{26} \) is already the simplest form.Thus, the exact value of \( \sin \left(\alpha - \frac{\pi}{3}\right) \) is \( \frac{-5 + 12\sqrt{3}}{26} \).

Key Concepts

Angle Subtraction Formula

Understanding the angle subtraction formula is crucial in trigonometry, especially when solving problems like finding the value of \( \sin(\alpha - \frac{\pi}{3}) \). The formula for sine difference is:

• \( \sin(a - b) = \sin a \cos b - \cos a \sin b \).

This formula allows you to express the sine of a difference of two angles in terms of their individual sines and cosines. It helps to break down complex trigonometric problems into more manageable calculations. In our exercise, \( a \) corresponds to \( \alpha \), and \( b \) is \( \frac{\pi}{3} \). This means we need to substitute into:

• \( \sin \alpha \cos \frac{\pi}{3} - \cos \alpha \sin \frac{\pi}{3} \).

Calculating these products and sums correctly is key to accurately determining the exact value of the trigonometric expression.

Quadrant Identification

Quadrant identification is essential when determining the sign and value of trigonometric functions. An angle's quadrant provides critical insights into the behavior of sine, cosine, and tangent. In the initial problem, we are given:

• \( \sin \alpha = -\frac{5}{13} \)
• \( \tan \alpha > 0 \)

The information indicates that \( \alpha \) is in the third quadrant. In this quadrant:

• Sine and cosine are negative,
• Tangent is positive (since both sine and cosine are negative).

Identifying the quadrant accurately helps with determining the correct signs of the trigonometric functions, ensuring the results are precise in subsequent calculations such as finding \( \cos \alpha \) and applying the angle subtraction formula.

Sine and Cosine Values

Accurately finding sine and cosine values within a particular quadrant is key to solving trigonometric problems. Given \( \sin \alpha = -\frac{5}{13} \), we need to determine \( \cos \alpha \) using the Pythagorean identity:

• \( \sin^2 \alpha + \cos^2 \alpha = 1 \).

Substituting the known sine value gives us:

• \( (-\frac{5}{13})^2 + \cos^2 \alpha = 1 \).
• \( \cos^2 \alpha = \frac{144}{169} \).

Since \( \alpha \) is in the third quadrant where cosine is negative, \( \cos \alpha = -\frac{12}{13} \). Understanding these calculations ensures you can use correct values in further trigonometric operations.

Simplifying Expressions

Simplifying expressions in trigonometry involves combining and reducing terms to find a cleaner form. Once the expression for \( \sin(\alpha - \frac{\pi}{3}) \) is derived using the angle subtraction formula, you get:

• \( \left(-\frac{5}{13}\right) \cdot \frac{1}{2} - \left(-\frac{12}{13}\right) \cdot \frac{\sqrt{3}}{2} \).

Following the calculations and simplifications, this expression results in:

• \( \frac{-5}{26} + \frac{12\sqrt{3}}{26} \).
• Final simplified form: \( \frac{-5 + 12\sqrt{3}}{26} \).

Simplifying such trigonometric expressions demands precision in arithmetic and algebraic manipulation, ensuring the final answer is both correct and in its simplest possible form.