Problem 53 - Precalculus Mathematics for Calculus

Question

Evaluate each expression under the given conditions. \(\sin 2 \theta ; \sin \theta=\frac{1}{7}, \theta\) in Quadrant II

Step-by-Step Solution

Answer

\( \sin 2\theta = -\frac{8\sqrt{3}}{49} \)

Step 1: Identify Relevant Formula

We need to evaluate \( \sin 2\theta \). The formula for the double angle of sine is \( \sin 2\theta = 2 \sin \theta \cos \theta \). We know \( \sin \theta = \frac{1}{7} \), but we need to find \( \cos \theta \).

Step 2: Use Pythagorean Identity

In Quadrant II, the cosine is negative. The Pythagorean identity states \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \sin \theta = \frac{1}{7} \) into the identity: \( \left(\frac{1}{7}\right)^2 + \cos^2 \theta = 1 \).

Step 3: Solve for \(\cos \theta\)

Calculate \( \frac{1}{7} = 0.142857 \), so \( \left(\frac{1}{7}\right)^2 = \frac{1}{49} \). Thus, \( \frac{1}{49} + \cos^2 \theta = 1 \). Solve this to get \( \cos^2 \theta = 1 - \frac{1}{49} = \frac{48}{49} \). So \( \cos \theta = -\sqrt{\frac{48}{49}} = -\frac{\sqrt{48}}{7} = -\frac{4\sqrt{3}}{7} \).

Step 4: Compute \(\sin 2\theta\)

Now substitute the values back into the double angle formula: \( \sin 2\theta = 2 \cdot \frac{1}{7} \cdot \left(-\frac{4\sqrt{3}}{7}\right) = -\frac{8\sqrt{3}}{49} \).

Key Concepts

Double Angle Formulas

The Double Angle Formulas are a set of essential tools in trigonometry, used to determine the trigonometric functions of angles that are twice as large as standard reference angles. For sine, the double angle formula is:

\[ \sin 2\theta = 2 \sin \theta \cos \theta \]Understanding this formula helps solve problems involving angles that are doubles of known angles. This formula means you need both the sine and cosine of the original angle to find the sine of the double angle.

In our example, since we are given \(\sin \theta = \frac{1}{7}\), we can directly use the formula once we have found the cosine of the angle \(\theta\). This formula is particularly useful because, in combination with other identities, it allows us to solve problems involving difficult trigonometric expressions.

Pythagorean Identity

The Pythagorean Identity is a crucial element in trigonometry, helping to relate the sine and cosine functions of an angle. The standard form of the identity is:

\[ \sin^2 \theta + \cos^2 \theta = 1 \]This identity states that for any angle \(\theta\), the square of the sine of \(\theta\) plus the square of the cosine of \(\theta\) always equals one. It effectively allows us to uncover one of the trigonometric values if we know the other.

In this example, knowing \(\sin \theta = \frac{1}{7}\) enables us to find \(\cos \theta\). By substituting \(\sin \theta\) into the identity, you solve for \(\cos^2 \theta\), yielding \(\cos \theta = -\frac{4\sqrt{3}}{7}\). Understanding such transformations is vital for simplifying trigonometric expressions efficiently.

Quadrant Analysis

Quadrant Analysis helps determine the signs of trigonometric functions based on the angle's position in the coordinate plane. The plane is divided into four quadrants, each affecting the sign of trigonometric values differently:

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive, cosine and tangent are negative.
• Quadrant III: Tangent is positive, sine and cosine are negative.
• Quadrant IV: Cosine is positive, sine and tangent are negative.

Since \(\theta\) is in Quadrant II, where sine values are positive and cosine values are negative, this corroborates why \(\cos \theta = -\frac{4\sqrt{3}}{7}\). Comprehending the quadrant system simplifies decision-making regarding signs when solving trigonometric problems, as acknowledging quadrant constraints is fundamental for correct evaluations.