Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Use identities to find (a) \(\sin 2 \theta\) and (b) \(\cos 2 \theta\) $$\sin \theta=\frac{4}{5} \text { and } \cos \theta<0$$

Step-by-Step Solution

Answer

(a) \(\sin 2\theta = -\frac{24}{25}\), (b) \(\cos 2\theta = -\frac{7}{25}\)

Step 1: Identify the Quadrant

We know \( \sin \theta = \frac{4}{5} \) and \( \cos \theta < 0 \). In which quadrant do both these conditions hold? In the second quadrant, \( \sin \theta \) is positive and \( \cos \theta \) is negative.

Step 2: Calculate \( \cos \theta \)

Using the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \sin \theta = \frac{4}{5} \) to get \( \left(\frac{4}{5}\right)^2 + \cos^2 \theta = 1 \). This simplifies to \( \frac{16}{25} + \cos^2 \theta = 1 \). Solving for \( \cos^2 \theta \), we get \( \cos^2 \theta = \frac{9}{25} \). Since \( \cos \theta < 0 \), \( \cos \theta = -\frac{3}{5} \).

Step 3: Use the Double Angle Identity to Find \( \sin 2\theta \)

The double angle identity for sine is \( \sin 2\theta = 2\sin \theta \cos \theta \). Substitute \( \sin \theta = \frac{4}{5} \) and \( \cos \theta = -\frac{3}{5} \): \( \sin 2\theta = 2 \times \frac{4}{5} \times -\frac{3}{5} = -\frac{24}{25} \).

Step 4: Use the Double Angle Identity to Find \( \cos 2\theta \)

The double angle identity for cosine is \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \). Substitute in the values: \( \cos 2\theta = \left(-\frac{3}{5}\right)^2 - \left(\frac{4}{5}\right)^2 = \frac{9}{25} - \frac{16}{25} = -\frac{7}{25} \).

Key Concepts

Double Angle Identities

The double angle identities are crucial tools in trigonometry, allowing us to express trigonometric functions of twice an angle in terms of the functions of the original angle. They are particularly useful for simplifying expressions and solving equations involving trigonometric functions.
The double angle identities for sine and cosine are:

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

To find \( \sin 2\theta \) and \( \cos 2\theta \), you can substitute the given values of \( \sin \theta \) and \( \cos \theta \) into these identities. For example, if \( \sin \theta = \frac{4}{5} \) and \( \cos \theta = -\frac{3}{5} \), then:

• Compute \( \sin 2\theta = 2 \times \frac{4}{5} \times -\frac{3}{5} = -\frac{24}{25} \)
• Find \( \cos 2\theta = \left(-\frac{3}{5}\right)^2 - \left(\frac{4}{5}\right)^2 = -\frac{7}{25} \)

These identities simplify the process by providing direct formulas, making calculations quicker and aiding in the understanding of more complex trigonometric concepts.

Pythagorean Identities

Pythagorean identities are fundamental relationships in trigonometry that express the interdependence between the sine and cosine of an angle based on the Pythagorean theorem. These identities are vital when you need to find missing values of trigonometric functions.The primary Pythagorean identity is:

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

This identity reflects the fact that for any angle, the square of the sine plus the square of the cosine will always equal one. Consider an example where \( \sin \theta = \frac{4}{5} \). You can use this identity to find the cosine value:

• Substitute \( \sin \theta = \frac{4}{5} \) into the identity: \( \left(\frac{4}{5}\right)^2 + \cos^2 \theta = 1 \)
• Simplify to get: \( \frac{16}{25} + \cos^2 \theta = 1 \)
• Solving for \( \cos^2 \theta \), you find \( \cos^2 \theta = \frac{9}{25} \)
• If \( \cos \theta < 0 \), then \( \cos \theta = -\frac{3}{5} \)

This identity is extremely useful for verifying solutions, simplifying expressions, and solving trigonometric equations.

Quadrants in Trigonometry

Understanding the quadrants in trigonometry is essential for determining the signs of trigonometric functions. The coordinate plane is divided into four quadrants, and the sign of the trigonometric function depends on which quadrant the angle's terminal side lies in.Here's a brief overview:

• **Quadrant I**: Both sine and cosine are positive.
• **Quadrant II**: Sine is positive, cosine is negative.
• **Quadrant III**: Both sine and cosine are negative.
• **Quadrant IV**: Sine is negative, cosine is positive.

In this exercise, since \( \sin \theta = \frac{4}{5} \) is positive and \( \cos \theta < 0 \) is negative, we conclude that \( \theta \) lies in Quadrant II. This insight helps in determining the correct signs when applying trigonometric identities such as the double angle identities or when using the Pythagorean identities.
Correctly identifying the quadrant is vital for predicting the behavior of the trigonometric function, aiding in solving problems accurately, and ensuring the solutions reflect the expected outcomes based on angle positions.