Problem 4 - Precalculus: Functions and Graphs

Question

Find the exact values of \(\sin 2 \theta, \cos 2 \theta,\) and \(\tan 2 \theta\) for the given values of \(\theta.\) $$\sin \theta=-\frac{4}{5} ; \quad 270^{\circ}<\theta<360^{\circ}$$

Step-by-Step Solution

Answer

\(\sin 2\theta = -\frac{24}{25}\), \(\cos 2\theta = -\frac{7}{25}\), and \(\tan 2\theta = \frac{24}{7}\).

Step 1: Determine the Quadrant

Since \(\sin \theta = -\frac{4}{5}\) and \(270^\circ < \theta < 360^\circ\), \(\theta\) is in the fourth quadrant, where sine is negative, cosine is positive, and tangent is negative.

Step 2: Calculate \(\cos \theta\) Using Pythagorean Identity

In any quadrant, \(\sin^2 \theta + \cos^2 \theta = 1\). Therefore, \(\cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(-\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25}\). Thus, \(\cos \theta = \frac{3}{5}\) because cosine is positive in the fourth quadrant.

Step 3: Calculate \(\sin 2 \theta\) Using Double Angle Formula

Use the formula \(\sin 2\theta = 2 \sin \theta \cos \theta\). Thus, \(\sin 2\theta = 2 \left(-\frac{4}{5}\right) \left(\frac{3}{5}\right) = -\frac{24}{25}\).

Step 4: Calculate \(\cos 2 \theta\) Using Double Angle Formula

Use the formula \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\). Using values from previous steps, \(\cos 2\theta = \frac{9}{25} - \frac{16}{25} = -\frac{7}{25}\).

Step 5: Calculate \(\tan 2 \theta\) Using \(\sin 2 \theta\) and \(\cos 2 \theta\)

Use the formula \(\tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta}\). Thus, \(\tan 2\theta = \frac{-\frac{24}{25}}{-\frac{7}{25}} = \frac{24}{7}\).

Key Concepts

Double Angle Formulas

Double angle formulas are essential tools in trigonometry. They help you find the trigonometric function values of angles that are double another given angle. For sine, cosine, and tangent, the double angle formulas are:

• For sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \)
• For cosine: \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
• For tangent: \( \tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta} \)

These formulas allow you to find values of these trigonometric functions by simply knowing \( \sin \theta \) and \( \cos \theta \). For example, in the original exercise, once you obtained \( \sin \theta = -\frac{4}{5} \) and \( \cos \theta = \frac{3}{5} \), calculating \( \sin 2\theta \), \( \cos 2\theta \), and \( \tan 2\theta \) became straightforward using these powerful formulas.

Pythagorean Identity

The Pythagorean identity is a cornerstone concept in trigonometry that states:\[\sin^2 \theta + \cos^2 \theta = 1\]This relation is derived from the Pythagorean theorem and holds true for any angle \( \theta \). It allows you to find one trigonometric function given the other. For example, if you know \( \sin \theta \), you can determine \( \cos \theta \) using:\[\cos^2 \theta = 1 - \sin^2 \theta\]In the original exercise, we used the Pythagorean identity to find \( \cos \theta \) from the given \( \sin \theta = -\frac{4}{5} \). By substituting, we found \( \cos^2 \theta = \frac{9}{25} \), leading to \( \cos \theta = \frac{3}{5} \), given that cosine is positive in the fourth quadrant.

Trigonometric Functions

Trigonometric functions are functions of angles that relate the angles of a triangle to the lengths of the sides of the triangle. The main trigonometric functions are:

• Sine \( (\sin) \)
• Cosine \( (\cos) \)
• Tangent \( (\tan) \)

These functions are crucial for solving various real-world problems involving periodic phenomena like sound and light waves. In trigonometry, knowing different expressions of these functions through identities or angle formulas is key to manipulating and solving equations efficiently. In particular, using these functions helped solve the exercise effectively. Once we had \( \sin \theta \) and calculated \( \cos \theta \), it was easy to apply additional formulas to find \( \sin 2\theta \), \( \cos 2\theta \), and \( \tan 2\theta \).

Quadrant Determination

Understanding quadrant determination is crucial when solving trigonometric equations. The coordinate system divides a circle into four quadrants, each affecting the sign of trigonometric functions:

• Quadrant I: \( \sin, \cos, \tan \) are positive.
• Quadrant II: \( \sin \) is positive, \( \cos \) and \( \tan \) are negative.
• Quadrant III: \( \tan \) is positive, \( \sin \) and \( \cos \) are negative.
• Quadrant IV: \( \cos \) is positive, \( \sin \) and \( \tan \) are negative.

The original exercise specifies \( 270^\circ < \theta < 360^\circ \), placing \( \theta \) in the fourth quadrant. Therefore, knowing the signs of functions is useful for deciding whether to take the positive or negative roots of values obtained from identities or solving equations. In our exercise, because \( \theta \) is in the fourth quadrant, it was concluded that \( \sin \theta = -\frac{4}{5} \) and \( \cos \theta = \frac{3}{5} \). This understanding is pivotal when using trigonometric functions and formulas.