Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-8,\) for each value of \(\theta,\) use double-angle formulas to find a. \(\sin 2 \theta,\) b. \(\cos 2 \theta,\) c. \(\tan 2 \theta .\) Show all work. $$ \theta=225^{\circ} $$

Step-by-Step Solution

Answer

For \( \theta = 225^{\circ} \), \( \sin 2\theta = \frac{1}{2} \), \( \cos 2\theta = 0 \), and \( \tan 2\theta \) is undefined.

Step 1: Convert Angle to Radians

To use trigonometric formulas effectively, we first convert the given angle \( \theta \) from degrees to radians. Since \( \theta = 225^{\circ} \), we use the conversion formula \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). So, \( 225^{\circ} = 225 \times \frac{\pi}{180} = \frac{5\pi}{4} \text{ radians} \).

Step 2: Use Double-Angle Formulas for Sin

Utilize the double-angle formula for sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \). First, find \( \sin \theta \) and \( \cos \theta \) for \( \theta = \frac{5\pi}{4} \). In the unit circle, \( \theta = 225^{\circ} \) corresponds to \( \sin \theta = -\frac{\sqrt{2}}{2} \) and \( \cos \theta = -\frac{\sqrt{2}}{2} \). Then, \( \sin 2\theta = 2(-\frac{\sqrt{2}}{2})(-\frac{\sqrt{2}}{2}) = \frac{1}{2} \).

Step 3: Use Double-Angle Formula for Cos

Use the double-angle formula for cosine: \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \). Since \( \cos \theta = -\frac{\sqrt{2}}{2} \) and \( \sin \theta = -\frac{\sqrt{2}}{2} \), substitute these values: \( \cos 2\theta = \left(-\frac{\sqrt{2}}{2}\right)^2 - \left(-\frac{\sqrt{2}}{2}\right)^2 = 0 \).

Step 4: Use Double-Angle Formula for Tan

Use the double-angle formula for tangent: \( \tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta} \). First, find \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1 \). Then substitute \( \tan \theta \): \( \tan 2\theta = \frac{2 \times 1}{1 - 1^2} = \text{undefined} \). This is because the denominator becomes zero owing to \( 1 - 1 = 0 \).

Key Concepts

Sine Double-Angle Formula

The sine double-angle formula is a key identity in trigonometry used to find the sine of a doubled angle given the sine and cosine of the original angle. The formula is expressed as:

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

To apply this formula effectively, it's important to know the values of \( \sin \theta \) and \( \cos \theta \). In our example, for \( \theta = 225^{\circ} \), we first convert \( \theta \) to radians as important groundwork for using trigonometric identities in calculus or higher-level mathematics.
Next, we calculate \( \sin \theta = -\frac{\sqrt{2}}{2} \) and \( \cos \theta = -\frac{\sqrt{2}}{2} \) based on their position on the unit circle. With these values in place, applying the sine double-angle formula leads us to:

• \[ \sin 2\theta = 2 \left(-\frac{\sqrt{2}}{2}\right) \left(-\frac{\sqrt{2}}{2}\right) = \frac{1}{2} \]

This demonstrates how the formula helps simplify calculations by using well-known trigonometric resources.

Cosine Double-Angle Formula

The cosine double-angle formula can be seen in different forms, each useful in various scenarios. The most common form is:

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

Using this formula allows us to find \( \cos 2\theta \) by substituting the known values of \( \sin \theta \) and \( \cos \theta \). For \( \theta = 225^{\circ} \), both \( \sin \theta \) and \( \cos \theta \) are \(-\frac{\sqrt{2}}{2}\). Substituting these into the formula gives us:

• \[ \cos 2\theta = \left(-\frac{\sqrt{2}}{2}\right)^2 - \left(-\frac{\sqrt{2}}{2}\right)^2 = 0 \]

This result occurs because the two squared values cancel each other. It's fascinating how these identities help break down seemingly complex trigonometric expressions into simple, calculable components.

Tangent Double-Angle Formula

The tangent double-angle formula efficiently handles the task of calculating \( \tan 2\theta \) by leveraging \( \tan \theta \):

• \( \tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta} \)

To use this formula, evaluate \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), which simplifies to \( 1 \) for \( \theta = 225^{\circ} \) (since both \( \sin \theta \) and \( \cos \theta \) are equal, their negative signs cancel).
However, substituting \( \tan \theta = 1 \) into the tangent double-angle formula results in a complication:

• \[ \tan 2\theta = \frac{2 \times 1}{1 - 1^2} = \text{undefined} \]

The result is undefined due to division by zero, highlighting a critical situation where angles lead to specific outcomes. Understanding these pitfalls prepares you for more complex trigonometric applications and equations.