Problem 88 - Algebra and Trigonometry

Question

find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator. $$ \sin \frac{\pi}{4} \cos 0-\sin \frac{\pi}{6} \cos \pi $$

Step-by-Step Solution

Answer

The exact value of the expression is \(\frac{\sqrt{2}}{2} + \frac{1}{2}\).

Step 1: Find the value of each trigonometric function

First, calculate the value of each trigonometric function for its given input. The sine and cosine functions have specific output values for the input angles \(\frac{\pi}{4}\), \(\frac{\pi}{6}\), 0 and \(\pi\). Following their respective ratios on the unit circle: \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\), \(\cos 0 = 1\), \(\sin \frac{\pi}{6} = \frac{1}{2}\), and \(\cos \pi = -1\).

Step 2: Substitute the values

Substitute the values obtained in step 1 into the original expression: \(\sin \frac{\pi}{4} \cos 0-\sin \frac{\pi}{6} \cos \pi = \frac{\sqrt{2}}{2} \cdot 1 - \frac{1}{2} \cdot -1\).

Step 3: Simplify the expression

The next step is to simplify the expression obtained in step 2. This yields: \(\frac{\sqrt{2}}{2} \cdot 1 - \frac{1}{2} \cdot -1 = \frac{\sqrt{2}}{2} + \frac{1}{2}\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It's a circle with a radius of 1, centered at the origin (0, 0) in the coordinate plane. The unit circle helps us visualize and understand the trigonometric functions.

By placing the angle's terminal side on this circle, the coordinates of the point where it intersects give us thecosine and sine of that angle:

• The x-coordinate represents the cosine.
• The y-coordinate represents the sine.

This circle allows us to quickly determine the exact values of trigonometric functions for common angles, such as \(\frac{\pi}{4}\), \(\frac{\pi}{6}\), 0, and \(\pi\). It simplifies calculations and provides a geometricinterpretation of the same.

Sine Function

The sine function is one of the primary trigonometric functions, and it's often associated with vertical movement.When looking at an angle on the unit circle, the sine value corresponds to the y-coordinate of the circle'sintersection.

For example:

• At \( \frac{\pi}{4} \), \( \sin \frac{\pi}{4} \) is equal to \( \frac{\sqrt{2}}{2} \).
• At \( \frac{\pi}{6} \), \( \sin \frac{\pi}{6} \) is equal to \( \frac{1}{2} \).

These specific outputs can be remembered or derived using the unit circle. The sine function is handy for solving problems involving triangles and wave patterns.

Cosine Function

Like the sine, the cosine function is essential in trigonometry, closely tied with horizontal movement. On the unitcircle, the cosine value of an angle gives the x-coordinate where the angle intersects the circle.

For particular angles:

• \( \cos 0 \) equals 1, representing the angle right at the positive x-axis.
• \( \cos \pi \) equals -1, placing the angle directly on the negative x-axis.

Understanding the cosine function and its exact values for standard angles gives us tools to solve trigonometricproblems efficiently. It's crucial for modeling cyclical patterns and understanding wave behavior.

Exact Values

Exact values in trigonometry don't need decimals or approximations when dealing with specific angles. Using the unitcircle, sine and cosine functions offer precise outputs for angles like \( \frac{\pi}{4} \), \( \frac{\pi}{6} \), 0, and \(\pi \).

The original expression \( \sin \frac{\pi}{4} \cos 0-\sin \frac{\pi}{6} \cos \pi \) simplifies seamlessly into exact numbers. By knowing:

• \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)
• \( \cos 0 = 1 \)
• \( \sin \frac{\pi}{6} = \frac{1}{2} \)
• \( \cos \pi = -1 \)

We directly substitute these to compute without any estimation, fully understanding the power of having exact values incalculations.