Problem 29 - Precalculus Mathematics for Calculus

Question

Evaluate the expression without using a calculator. $$\left(\cos 30^{\circ}\right)^{2}-\left(\sin 30^{\circ}\right)^{2}$$

Step-by-Step Solution

Answer

The evaluated expression is \(\frac{1}{2}\).

Step 1: Recall Trigonometric Values

First, let's recall the values of the trigonometric functions for \(30^{\circ}\). We have \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\) and \(\sin 30^{\circ} = \frac{1}{2}\).

Step 2: Substitute Values into the Expression

Substitute the trigonometric values into the expression \[(\cos 30^{\circ})^2 - (\sin 30^{\circ})^2 = \left(\frac{\sqrt{3}}{2}\right)^2 - \left(\frac{1}{2}\right)^2\].

Step 3: Square the Trigonometric Values

Compute the squares: \[\left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\quad \text{and}\quad \left(\frac{1}{2}\right)^2 = \frac{1}{4}\].

Step 4: Subtract the Squares

Subtract the squared values to get the final answer: \[\frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2}\].

Key Concepts

Trigonometric Functions

Trigonometric functions are a fundamental part of mathematics, especially when dealing with triangles and circular arcs. The most commonly used trigonometric functions include sine, cosine, and tangent. These functions are essentially ratios of different sides of a right triangle.

• Sine (sin) is the ratio of the opposite side to the hypotenuse.
• Cosine (cos) is the ratio of the adjacent side to the hypotenuse.
• Tangent (tan) is the ratio of the opposite side to the adjacent side.

These functions are not only used in geometry but also in fields like physics, engineering, and computer science. Understanding these functions can provide insights into phenomena ranging from sound waves to oscillatory movements.

Cosine and Sine Values

To solve problems involving trigonometric functions, it's essential to know the common values for sine and cosine, especially for well-known angles like 30°, 45°, 60°, etc.
For the angle of 30°:

• The value of \(\cos 30^\circ\) is \(\frac{\sqrt{3}}{2}\).
• The value of \(\sin 30^\circ\) is \(\frac{1}{2}\).

These values are derived from special triangles, such as the 30°-60°-90° triangle, where the length of the sides are in the ratio of 1:√3:2. Remembering these values can greatly simplify the task of working with trigonometric expressions.

Angle in Degrees

Angles can be measured in different units, mainly degrees and radians. In most elementary trigonometry, angles are given in degrees.

• One full revolution around a circle is 360 degrees.
• Common angle measures include 30°, 45°, and 60°, vital in trigonometry.

Understanding these angle measures helps in evaluating trigonometric expressions because it tells us where on the unit circle the angles point to. Knowing this can help recall specific sine and cosine values useful for calculations.

Expression Evaluation

Evaluating trigonometric expressions involves several steps, often requiring substitution of known values and performing basic arithmetic operations.
To evaluate the expression \(\left(\cos 30^\circ\right)^{2} - \left(\sin 30^\circ\right)^{2}\), follow these steps:

• First, substitute \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) and \(\sin 30^\circ = \frac{1}{2}\).
• Next, square these values: \(\left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\) and \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\).
• Finally, subtract the squared values: \(\frac{3}{4} - \frac{1}{4} = \frac{1}{2}\).

Approaching a problem step by step and employing known trigonometric identities simplify the process, allowing you to evaluate expressions without a calculator.