Problem 29 - Algebra and Trigonometry

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 expression evaluates to \( \frac{1}{2} \).

Step 1: Identify Trigonometric Values

First, determine the trigonometric values of \( \cos 30^{\circ} \) and \( \sin 30^{\circ} \). From trigonometric tables or knowledge, we know \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \) and \( \sin 30^{\circ} = \frac{1}{2} \).

Step 2: Substitute Values into Expression

Replace \( \cos 30^{\circ} \) and \( \sin 30^{\circ} \) in the expression \( \left(\cos 30^{\circ}\right)^{2}-\left(\sin 30^{\circ}\right)^{2} \) with their corresponding values. The expression becomes \( \left(\frac{\sqrt{3}}{2}\right)^{2} - \left(\frac{1}{2}\right)^{2} \).

Step 3: Compute Squares

Calculate \( \left(\frac{\sqrt{3}}{2}\right)^{2} \) and \( \left(\frac{1}{2}\right)^{2} \). This results in \( \frac{3}{4} \) for \( \cos^2 30^{\circ} \) and \( \frac{1}{4} \) for \( \sin^2 30^{\circ} \).

Step 4: Perform Subtraction

Subtract the computed \( \sin^2 30^{\circ} \) from \( \cos^2 30^{\circ} \). That is, calculate \( \frac{3}{4} - \frac{1}{4} \), which simplifies to \( \frac{2}{4} = \frac{1}{2} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental equations in trigonometry that hold true for any value of the involved variables. They are vital in simplifying and solving trigonometric expressions. In the given exercise, the expression \( \cos^2 30^{\circ} - \sin^2 30^{\circ} \) involves one such identity known as the "cosine and sine squared identity".

• The identity \( \cos^2 \theta - \sin^2 \theta = \cos(2\theta) \) is a derivative of the double angle formulas.
• We can use this identity to transform complex trigonometric expressions into simpler forms. In our exercise, utilizing this identity directly gives us \( \cos(60^{\circ}) \), which equals \( \frac{1}{2} \).

Trigonometric identities allow us to transform expressions without performing exhaustive calculations, making problem-solving more efficient.

Trigonometric Values

Understanding and memorizing basic trigonometric values is crucial for evaluating expressions without a calculator. These values are typically associated with special angles, such as \(0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ},\) and \(90^{\circ}\).
Specific trigonometric values for these angles include:

• \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\)
• \(\sin 30^{\circ} = \frac{1}{2}\)

These values stem from geometric principles within a unit circle or right-angle triangles, helping simplify the calculation of trigonometric expressions. Knowing such values by heart aids in evaluating expressions like those seen in the exercise, streamlining the process.

Cosine and Sine

Cosine and sine are foundational trigonometric functions defining the ratios of the sides of a right-angled triangle in relation to its angles. In trigonometric expressions, cosine and sine values are used to describe rotational and wave phenomena.

• Cosine: Represents the length of the side adjacent to the angle divided by the hypotenuse.
• Sine: Represents the length of the side opposite the angle divided by the hypotenuse.

In the expression \( \left(\cos 30^{\circ}\right)^2 - \left(\sin 30^{\circ}\right)^2 \), the cosine and sine values provide a measure of the angle's horizontal and vertical components, respectively.
By substituting \( \cos 30^{\circ} \) and \( \sin 30^{\circ} \) into the equation, we perform algebraic operations to find the result. This understanding is essential for working with any trigonometric problems where these functions are present, facilitating deeper comprehension of their applications.