Problem 53 - Algebra and Trigonometry

Question

If \(\theta=\pi / 3,\) find the value of each expression. \(\begin{array}{ll}{\text { (a) } \sin 2 \theta,} & {2 \sin \theta} & {\text { (b) } \sin \frac{1}{2} \theta, \quad \frac{1}{2} \sin \theta} \\ {\text { (c) } \sin ^{2} \theta,} & {\sin \left(\theta^{2}\right)}\end{array}\)

Step-by-Step Solution

Answer

(a) \(\sin 2\theta = \frac{\sqrt{3}}{2}, 2\sin\theta = \sqrt{3}\); (b) \(\sin \frac{1}{2}\theta = \frac{1}{2}, \frac{1}{2}\sin\theta = \frac{\sqrt{3}}{4}\); (c) \(\sin^2 \theta = \frac{3}{4}\).

Step 1: Simplify and Calculate 2θ

First, calculate the double angle using the formula for the sine of a double angle: \( \sin 2\theta = 2 \sin \theta \cos \theta \). Since \( \theta = \frac{\pi}{3} \), we know \( \sin \theta = \frac{\sqrt{3}}{2} \) and \( \cos \theta = \frac{1}{2} \). So, \( \sin 2\theta = 2 \times \frac{\sqrt{3}}{2} \times \frac{1}{2} = \frac{\sqrt{3}}{2} \).

Step 2: Comparison of 2 sin θ

Calculate \( 2\sin\theta = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3} \). Compare \( \sin 2\theta = \frac{\sqrt{3}}{2} \) and \( 2\sin\theta = \sqrt{3} \).

Step 3: Calculate \(\sin \frac{1}{2}\theta\)

Use the half-angle formula: \( \sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}} \). With \( \cos \theta = \frac{1}{2} \), \( \sin \frac{\theta}{2} = \sqrt{\frac{1 - \frac{1}{2}}{2}} = \sqrt{\frac{1}{4}} = \frac{1}{2} \).

Step 4: Comparison of \(\frac{1}{2} \sin \theta\)

Compute \( \frac{1}{2} \sin \theta = \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4} \). Compare \( \sin \frac{1}{2} \theta = \frac{1}{2} \) and \( \frac{1}{2} \sin \theta = \frac{\sqrt{3}}{4} \).

Step 5: Calculate \( \sin^2 \theta \)

Square \( \sin \theta \): \( (\sin \theta)^2 = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \).

Step 6: Analyze \( \sin(\theta^2) \)

Since \( \theta = \frac{\pi}{3} \), \( \theta^2 = \left(\frac{\pi}{3}\right)^2 = \frac{\pi^2}{9} \). The function \( \sin(\theta^2) \) might not have a simple angle result. This requires computation not based on known reference angles, thus it typically involves approximation or further tools not specified in the context.

Key Concepts

Double Angle Formula

The double angle formula is a valuable tool used to express trigonometric functions of double angles in terms of single angles. Specifically, the formula for sine double angles is given by:\[\sin 2\theta = 2 \sin \theta \cos \theta\]This formula is beneficial when solving problems involving functions like \(\sin 2\theta\), where direct calculation might be challenging. Instead, by knowing the sine and cosine of the angle \(\theta\), you can easily find \(\sin 2\theta\). It is particularly useful in problems where the angle \(\theta\) is a standard angle like \(\pi / 3\). - For example, if \(\theta = \pi / 3\), using the known values \(\sin \theta = \frac{\sqrt{3}}{2}\) and \(\cos \theta = \frac{1}{2}\), the calculation becomes straightforward:
\[\sin 2\theta = 2 \times \frac{\sqrt{3}}{2} \times \frac{1}{2} = \frac{\sqrt{3}}{2}\]The double angle formula not only simplifies calculations but also aids in verifying trigonometric expressions.

Half Angle Formula

The half angle formula allows us to calculate trigonometric functions for half angles. It is particularly useful when processing expressions like \(\sin \frac{\theta}{2}\) where direct computation may not be feasible. The sine half-angle formula is as follows:\[\sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}}\]This expression is derived from the known identity for cosine of double angles, and it helps us find a trigonometric value for half of an angle from its cosine value.- To illustrate, assume \(\theta = \pi / 3\) giving \(\cos \theta = \frac{1}{2}\). Substituting this into the half-angle formula, we get:
\[\sin \frac{\theta}{2} = \sqrt{\frac{1 - \frac{1}{2}}{2}} = \sqrt{\frac{1}{4}} = \frac{1}{2}\]The half angle formulas can make trigonometric calculations more efficient and provide checks for accuracy by simplifying complex expressions.

Sine Function

The sine function is a fundamental part of the trigonometric suite of functions. It is periodic and takes real values on a continuous interval. For a given angle \(\theta\), \(\sin \theta\) represents the y-coordinate of a point on the unit circle where the terminal side of the angle intersects it.- A notable property of the sine function is its symmetry, being an odd function defined by \(\sin(-\theta) = -\sin(\theta)\). - Calculating the square of the sine function, \(\sin^2 \theta\), is often encountered in trigonometric identities and can simplify further algebraic manipulation.For instance, when \(\theta = \pi / 3\), the sine is known to be \(\sin \theta = \frac{\sqrt{3}}{2}\). Therefore:
\[(\sin \theta)^2 = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\]Understanding how to manipulate the sine function is essential for comprehensively analyzing trigonometric identities and functions.

Cosine Function

The cosine function, like the sine function, is a foundational trigonometric function. It represents the x-coordinate on a unit circle of the point where an angle’s terminal side crosses. For any angle \(\theta\), \(\cos \theta\) is cyclical, producing repetitive values over its defined interval.The cosine function features prominently in functions such as the half-angle and double angle identities. It is an even function, meaning \(\cos(-\theta) = \cos(\theta)\), which adds to its mathematical elegance and utility in problem-solving.- Taking an example of \(\theta = \pi / 3\), the cosine value is \(\cos \theta = \frac{1}{2}\). If you were to compute the expression \(\sin(\theta)^2 + \cos(\theta)^2\), you would find:
\[\frac{3}{4} + \frac{1}{4} = 1\]This highlights the Pythagorean identity, confirming that sine and cosine are crucial elements in trigonometric computations.