Problem 6 - 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. $$ \theta=\frac{\pi}{4} $$

Step-by-Step Solution

Answer

\( \sin 2\theta = 1 \), \( \cos 2\theta = 0 \), \( \tan 2\theta \) is undefined.

Step 1: Identify the Double-Angle Formulas

To solve for \( \sin 2\theta \), \( \cos 2\theta \), and \( \tan 2\theta \), we need to use the double-angle formulas: \( \sin 2\theta = 2 \sin \theta \cos \theta \), \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \), and \( \tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta} \).

Step 2: Compute \( \sin \theta \) and \( \cos \theta \)

Since \(\theta = \frac{\pi}{4}\), we know from trigonometric identities that \( \sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

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

Apply the formula \( \sin 2\theta = 2 \sin \theta \cos \theta \). Substituting \( \sin \theta = \frac{\sqrt{2}}{2} \) and \( \cos \theta = \frac{\sqrt{2}}{2} \), we get \( \sin 2\theta = 2 \times \frac{\sqrt{2}}{2} \times \frac{\sqrt{2}}{2} = 2 \times \frac{1}{2} = 1 \).

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

Use the double-angle formula \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \). Substituting \( \cos \theta = \frac{\sqrt{2}}{2} \) and \( \sin \theta = \frac{\sqrt{2}}{2} \), we have \( \cos 2\theta = \left(\frac{\sqrt{2}}{2}\right)^2 - \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{1}{2} - \frac{1}{2} = 0 \).

Step 5: Calculate \( \tan 2\theta \)

For \( \tan 2\theta \), use the formula \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \). With \( \tan \frac{\pi}{4} = 1 \), it results in \( \tan 2\theta = \frac{2 \times 1}{1 - 1^2} = \frac{2}{0} \). Since dividing by zero is undefined, \( \tan 2\theta \) is not defined.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics that create connections between different trigonometric functions. These identities simplify complex equations and are particularly useful when dealing with angles in various applications, such as physics and engineering. Trigonometric identities involve:

• Basic identities like the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Angle sum and difference identities
• Double-angle and half-angle identities

By using these identities, you can easily convert between functions and solve complicated trigonometric equations with greater efficiency. Understanding trigonometric identities offers a strong foundation for mastering trigonometry and calculus.

sin 2 theta

The double-angle formula for sine allows us to express \( \sin 2\theta \) as a function of \( \sin \theta \) and \( \cos \theta \):\[\sin 2\theta = 2 \sin \theta \cos \theta\]This formula is incredibly helpful, as it simplifies expressions where the angle is doubled, which is common in wave and oscillation problems.**Example Calculation:**
Suppose \( \theta = \frac{\pi}{4} \). We know from basic trigonometric values that:

• \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)
• \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)

Substituting these values into the double-angle formula gives:\[\sin 2\theta = 2 \times \frac{\sqrt{2}}{2} \times \frac{\sqrt{2}}{2} = 1\]Thus, \( \sin 2\theta = 1 \) when \( \theta = \frac{\pi}{4} \).

cos 2 theta

The cosine double-angle formula provides a way to calculate \( \cos 2\theta \) using only sine and cosine of the original angle \( \theta \):\[\cos 2\theta = \cos^2 \theta - \sin^2 \theta\]It’s versatile as it can also be written in other equivalent forms, such as:

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

These forms are useful depending on which function, sine or cosine, is available or easier to use.**Example Calculation:**
For \( \theta = \frac{\pi}{4} \):

• \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)
• \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)

Plug into the formula:\[\cos 2\theta = \left( \frac{\sqrt{2}}{2} \right)^2 - \left( \frac{\sqrt{2}}{2} \right)^2 = \frac{1}{2} - \frac{1}{2} = 0\]So, \( \cos 2\theta = 0 \) when \( \theta = \frac{\pi}{4} \).

tan 2 theta

To find \( \tan 2\theta \), we use the double-angle formula for tangent:\[\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}\]This formula helps compute the tangent of a doubled angle based purely on the tangent of the single angle. It can be highly useful in situations involving angle transformations.**Example Calculation:**
With \( \theta = \frac{\pi}{4} \), we have:

• \( \tan \frac{\pi}{4} = 1 \)

Substituting into the formula:\[\tan 2\theta = \frac{2 \cdot 1}{1 - 1^2} = \frac{2}{0}\]This result is undefined because division by zero is not possible. Hence, \( \tan 2\theta \) is undefined when \( \theta = \frac{\pi}{4} \). This typically marks a point where the tangent curve undergoes a vertical asymptote, illustrating the periodic nature of the tangent function.