Problem 1 - 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 \(\theta.\) $$\cos \theta=\frac{3}{5} ; \quad 0^{\circ}<\theta<90^{\circ}$$

Step-by-Step Solution

Answer

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

Step 1: Understand Given Information

We are given that \( \cos \theta = \frac{3}{5} \) and \( 0^{\circ} < \theta < 90^{\circ} \). This tells us \( \theta \) is in the first quadrant where all trigonometric functions are positive.

Step 2: Find \( \sin \theta \) Using Pythagorean Identity

Since \( 0^{\circ} < \theta < 90^{\circ} \), we use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \).\[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{3}{5}\right)^2 = \frac{16}{25} \]Then \( \sin \theta = \sqrt{\frac{16}{25}} = \frac{4}{5} \) because \( \sin \theta \) is positive in the first quadrant.

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

The double angle formula for sine is \( \sin 2\theta = 2 \sin \theta \cos \theta \).\[ \sin 2\theta = 2 \times \frac{4}{5} \times \frac{3}{5} = \frac{24}{25} \]

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

The double angle formula for cosine is \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \).\[ \cos 2\theta = \left(\frac{3}{5}\right)^2 - \left(\frac{4}{5}\right)^2 = \frac{9}{25} - \frac{16}{25} = -\frac{7}{25} \]

Step 5: Calculate \( \tan 2 \theta \) using the Identity

Use the identity \( \tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta} \).\[ \tan 2\theta = \frac{\frac{24}{25}}{-\frac{7}{25}} = -\frac{24}{7} \]

Key Concepts

First Quadrant Angles

In trigonometry, angles are often divided into four quadrants. The first quadrant, where \(0^\circ < \theta < 90^\circ\), holds special significance for students. This is because all trigonometric functions like sine, cosine, and tangent are positive in this first quadrant. This positivity translates into simplicity, as there's no need to worry about negative values which often occur in other quadrants. Students should keep in mind that whenever dealing with trigonometric values within this range, as in our given exercise, positive results should be expected for each of the primary trigonometric ratios. Here, it's crucial to understand that when given \(\cos \theta = \frac{3}{5}\) and knowing that the angle is in the first quadrant, we can directly work with positive values for computations. This is especially useful for simplifying calculations and checking the final results.

Double Angle Formulas

Double angle formulas are fundamental tools in trigonometry. These formulas enable us to find the trigonometric function values for expressions like \(2\theta\). There are key formulas for sine and cosine:

• The formula for sine is \(\sin 2\theta = 2 \sin \theta \cos \theta\)
• The formula for cosine is \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\)

These formulas reduce complex expressions into simpler terms, transforming double angles into known angles, making it easier to solve problems. In the exercise, once we know \(\sin \theta\) and \(\cos \theta\), applying these formulas becomes straightforward and gives accurate results for both sine and cosine of the double angle without the need for more complex techniques.

Pythagorean Identity

The Pythagorean identity is a foundation in trigonometry that correlates the sine and cosine of a given angle. It states that \(\sin^2 \theta + \cos^2 \theta = 1\). This identity helps find unknown trigonometric values when one is already known. Specifically, once we know \(\cos \theta\), we can compute \(\sin \theta\) by rearranging the identity:\[ \sin^2 \theta = 1 - \cos^2 \theta \]This identity assures that knowing one function value automatically enables the calculation of the other, which is invaluable in exercises like the one provided. It establishes a ‘triangle’ of relationships between sine, cosine, and the concept of unity (1), making it a powerful tool in trigonometric analysis. For students, recognizing and correctly applying the Pythagorean identity enhances their problem-solving capabilities and deepens their understanding of the interdependence of trigonometric functions.