Problem 9 - Precalculus Essentials

Question

In Exercises \(7-14,\) use the given information to find the exact value of each of the following: a. \(\sin 2 \theta\) b. \(\cos 2 \theta\) c. \(\tan 2 \theta\) \(\cos \theta=\frac{24}{25}, \theta\) lies in quadrant IV.

Step-by-Step Solution

Answer

a. \(\sin 2 \theta = -\frac{1344}{625}\) b. \(\cos 2 \theta = \frac{336}{625}\) c. \(\tan 2 \theta = -4\).

Step 1: Calculate \(\sin \theta\)

Given \(\cos \theta=\frac{24}{25}\) we calculate \(\sin \theta\) using the Pythagorean identity \(\cos^2\theta + \sin^2\theta = 1\). Solving this equation for \(\sin\theta\) results in \(\sin\theta = \pm \sqrt{1 - \cos^2\theta} = \pm \sqrt{1 - (\frac{24}{25})^2} = \pm \frac{7}{25}\). Since \(\theta\) lies in the fourth quadrant, where \(\sin \theta\) is negative, we choose \(\sin \theta = -\frac{7}{25}\).

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

We use the double angle formula \(\sin 2 \theta = 2 \sin \theta \cos \theta\). Substituting the values of \(\sin \theta\) and \(\cos \theta\) gives \(\sin 2 \theta = 2 * -\frac{7}{25} * \frac{24}{25} = -\frac{1344}{625}\).

Step 3: Calculate \(\cos 2 \theta\)

We use the double angle formula \(\cos 2 \theta = \cos^2 \theta - \sin^2 \theta\). Substituting the values of \(\sin \theta\) and \(\cos \theta\) gives \(\cos 2 \theta = (\frac{24}{25})^2 - (-\frac{7}{25})^2 = \frac{336}{625}\).

Step 4: Calculate \(\tan 2 \theta\)

Using the formula \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), we calculate \(\tan 2 \theta = \frac{\sin 2 \theta}{\cos 2 \theta} = \frac{-\frac{1344}{625}}{\frac{336}{625}} = -4\).

Key Concepts

Double Angle Formulas

Double angle formulas are key tools in trigonometry for simplifying expressions involving trigonometric functions of double angles. They allow us to express trigonometric functions like \(\sin 2\theta\), \(\cos 2\theta\), and \(\tan 2\theta\) in terms of \(\sin \theta\) and \(\cos \theta\). These formulas help in many applications, such as simplifying expressions, solving equations, and even analyzing oscillatory systems.

• For sine: \(\sin 2\theta = 2 \sin \theta \cos \theta\).
• For cosine: \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\), which can also be expressed as \(2\cos^2\theta - 1\) or \(1 - 2\sin^2\theta\).
• For tangent: \(\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}\).

In our exercise, these formulas help us find the values of \(\sin 2\theta\), \(\cos 2\theta\), and \(\tan 2\theta\) given values of \(\sin \theta\) and \(\cos \theta\). This understanding is essential in building a strong trigonometry foundation.

Trigonometric Quadrants

The concept of trigonometric quadrants is crucial when solving trigonometric problems, as it determines the sign of trigonometric functions. The coordinate plane is divided into four quadrants, labeled I, II, III, and IV.

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

In our exercise, \(\theta\) lies in the fourth quadrant, indicating \(\sin\theta\) is negative and \(\cos\theta\) is positive. This information is critical in selecting the correct signs when determining the value of trigonometric functions.

Pythagorean Identity

The Pythagorean identity is one of the most fundamental identities in trigonometry. It states that for any angle \(\theta\), the square of the sine function plus the square of the cosine function is always equal to one: \(\sin^2 \theta + \cos^2 \theta = 1\). This identity is a reflection of the Pythagorean theorem applied to the unit circle where the hypotenuse is 1.
This identity is particularly powerful in transforming and simplifying expressions or solving equations where one trigonometric function is given, allowing you to find the other.
In the provided exercise, we used the Pythagorean identity to express \(\sin\theta\) in terms of \(\cos\theta\). Given that \(\cos\theta = \frac{24}{25}\), we plugged it into the identity to find \(\sin\theta = -\frac{7}{25}\), considering that \(\theta\) is in the fourth quadrant. This approach is essential for accurately solving trigonometric problems.