Problem 15 - A Graphical Approach to Precalculus with Limits

Question

Use identities to find (a) \(\sin 2 \theta\) and (b) \(\cos 2 \theta\) $$\sin \theta=-\frac{\sqrt{5}}{7} \text { and } \cos \theta>0$$

Step-by-Step Solution

Answer

\(\sin 2 \theta = -\frac{4 \sqrt{55}}{49}\), \(\cos 2 \theta = \frac{39}{49}\).

Step 1: Use Pythagorean Identity

First, identify the unknown trigonometric function using the Pythagorean identity. Since \(\sin \theta = -\frac{\sqrt{5}}{7}\), use the identity \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute \(\sin \theta\): \[(-\frac{\sqrt{5}}{7})^2 + \cos^2 \theta = 1\]\[\frac{5}{49} + \cos^2 \theta = 1\]Solve for \(\cos^2 \theta\):\[\cos^2 \theta = \frac{44}{49}\]

Step 2: Determine Sign of \(\cos \theta\)

Given that \(\cos \theta > 0\), take the positive square root of \(\cos^2 \theta\):\[\cos \theta = \frac{\sqrt{44}}{7} = \frac{2\sqrt{11}}{7}\]

Step 3: Find \(\sin 2 \theta\) using Double Angle Identity

Use the double angle identity for sine: \(\sin 2 \theta = 2 \sin \theta \cos \theta\). Substitute the known values:\[\sin 2 \theta = 2 \left(-\frac{\sqrt{5}}{7}\right) \left(\frac{2\sqrt{11}}{7}\right)\]Calculate the expression:\[\sin 2 \theta = -\frac{4 \sqrt{55}}{49}\]

Step 4: Find \(\cos 2 \theta\) using Double Angle Identity

Use the double angle identity for cosine: \(\cos 2 \theta = \cos^2 \theta - \sin^2 \theta\). Substitute the known values:\[\cos 2 \theta = \frac{44}{49} - \frac{5}{49}\]Calculate the expression:\[\cos 2 \theta = \frac{39}{49}\]

Key Concepts

Double Angle Identity

To understand the Double Angle Identity, it's helpful to see how it connects different angles in trigonometric functions. The Double Angle Identity is a trigonometric formula that expresses functions like sine, cosine, and tangent in terms of angles that are twice the original angle. For sine and cosine, the identities are:

• \(\sin 2\theta = 2 \sin \theta \cos \theta\)
• \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\)

Using these identities simplifies calculations when you know the sine and cosine of an angle, making it easy to find the double angle values.

By substituting known values, like in the original exercise, you can quickly determine \(\sin 2\theta\) and \(\cos 2\theta\). These identities are especially useful in solving trigonometric equations and simplifying expressions.

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry that relates the squares of sine and cosine functions to 1. The identity is expressed as:

• \(\sin^2 \theta + \cos^2 \theta = 1\)

This equation is derived from the Pythagorean theorem and is essential for finding unknown trigonometric values when one is given. For example, in the exercise, knowing \(\sin \theta = -\frac{\sqrt{5}}{7}\) allows us to find \(\cos^2 \theta\) by substituting this value into the identity.

Rearranging the equation helps in solving for \(\cos^2 \theta\): \[\cos^2 \theta = 1 - \sin^2 \theta\]

This step is crucial because, from \(\cos^2 \theta\), you can determine \(\cos \theta\) by considering the sign based on the given conditions, such as \(\cos \theta > 0\). The Pythagorean Identity provides a reliable foundation for solving many trigonometric problems.

Sine and Cosine Functions

Sine and cosine are two fundamental trigonometric functions that describe the relationships in a right triangle. The sine of an angle \(\theta\) is the ratio of the length of the opposite side to the hypotenuse. Similarly, the cosine is the ratio of the adjacent side's length to the hypotenuse. These basic definitions lead to other trigonometric identities that simplify calculations.

In the context of the exercise, observing that \(\sin \theta = -\frac{\sqrt{5}}{7}\) gives insight into which quadrant of the circle \(\theta\) lies, guided by the fact that sine is negative and cosine positive. This indicates \(\theta\) is in the fourth quadrant, where cosine values are positive, and sine values are negative.

Understanding the nature of sine and cosine facilitates solutions that involve manipulating these functions through various identities and equations. With the Double Angle and Pythagorean Identities, you are equipped to solve complex trigonometric challenges using insights into the behavior of these primary functions.