Problem 70 - Algebra and Trigonometry

Question

Express the exact value of each function as a single fraction. Do not use a calculator. $$ \text { If } f(\theta)=2 \sin \theta-\sin \frac{\theta}{2}, \text { find } f\left(\frac{\pi}{3}\right) $$

Step-by-Step Solution

Answer

The exact value of the function for \( \theta = \frac{\pi}{3} \) is \( \sqrt{3} - \frac{1}{2} \)

Step 1: Substitute the given theta value

Input \( \theta = \frac{\pi}{3} \) into the equation to get \( f(\frac{\pi}{3})=2 \sin \frac{\pi}{3}-\sin \frac{\pi}{6} \)

Step 2: Simplify the expression using trigonometric identities

Recall that \( \sin \frac{\pi}{3}= \frac{\sqrt{3}}{2} \) and \( \sin \frac{\pi}{6}= \frac{1}{2} \). Substituting these values in gives \( f(\frac{\pi}{3}) = 2(\frac{\sqrt{3}}{2}) - \frac{1}{2} \)

Step 3: Further Simplify the expression

Simplify the expression to get the final answer. \[ f(\frac{\pi}{3}) = \sqrt{3} - \frac{1}{2} \]

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that relate various trigonometric functions to one another. These identities are crucial in solving complex trigonometry problems without the need for a calculator. One of the fundamental trigonometric identities is the Pythagorean Identity, which connects the sine and cosine functions: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] Other important identities include the angle sum and difference identities, double angle formulas, and half-angle formulas. In the example exercise, using the identities to find the exact values of \( \sin\frac{\pi}{3} \) and \( \sin\frac{\pi}{6} \) is a key step in simplifying the function \( f(\theta) \).
Understanding and memorizing common trigonometric identities are instrumental not just for academic exercises but also for applications in physics and engineering, where trigonometric functions describe phenomena like waves and oscillations.

Sine Function

The sine function, often abbreviated as \( \sin \), is one of the primary trigonometric functions. It represents the ratio of the length of the side of the triangle opposite the angle to the length of the triangle's hypotenuse in a right-angled triangle. More generally, in the unit circle framework, the value of the sine function for any given angle is the y-coordinate of the arc endpoint on the unit circle. This allows us to define the sine function for any real number, extending the right triangle definition to a broader range of angles, including negative angles and angles greater than \( 2\pi \).
For the angle \( \theta = \frac{\pi}{6} \) (or 30 degrees), the sine value is \( \frac{1}{2} \), and for \( \theta = \frac{\pi}{3} \) (or 60 degrees), it is \( \frac{\sqrt{3}}{2} \) as used in the exercise. The utility of the sine function extends from purely mathematical fields to describing periodic phenomena in the natural world, such as the motion of waves.

Exact Trigonometric Values

Exact trigonometric values refer to the precise fractional, radical, or integer values of trigonometric functions for specific angles. These values are preferred in mathematical proofs and certain applications where approximate values (like those given by a calculator) could lead to errors. Many standard angles, such as \( 0 \), \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), and \( \frac{\pi}{2} \), have known exact values for the sine, cosine, and tangent functions.
For example, the exact value of \( \sin\frac{\pi}{3} \) is \( \frac{\sqrt{3}}{2} \) and not an approximate decimal. Learning these values assists in quickly simplifying trigonometric expressions and solving trigonometric equations effectively, as demonstrated in the given exercise. Obtaining an exact value, as opposed to a decimal approximation, is also essential in subjects that require a high degree of precision like engineering, physics, and computer science.