Problem 27 - 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: Recall Trigonometric Values

Recall the trigonometric values for cosine and sine of 30 degrees: \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \) and \( \sin 30^{\circ} = \frac{1}{2} \).

Step 2: Square the Trigonometric Values

Calculate the square of \( \cos 30^{\circ} \) and \( \sin 30^{\circ} \): \[(\cos 30^{\circ})^2 = \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4}\] and \[(\sin 30^{\circ})^2 = \left( \frac{1}{2} \right)^2 = \frac{1}{4}.\]

Step 3: Substitute and Simplify the Expression

Substitute the squared values back into the expression and simplify: \[\left( \cos 30^{\circ} \right)^2 - \left( \sin 30^{\circ} \right)^2 = \frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2}.\]

Key Concepts

Trigonometric Values

Trigonometric values are specific values of sine, cosine, and other trigonometric functions for common angles, such as 30°, 45°, and 60°. These values are often memorized or derived to solve problems efficiently without calculators. For 30°, the trigonometric values are:

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

These values are widely used because they frequently appear in geometric and trigonometric problems.Remembering these common values helps simplify expressions and solve equations more easily. When you encounter problems asking for evaluations of trigonometric expressions, having these values at your fingertips can prevent a lot of unnecessary computation.

Cosine and Sine

Cosine and sine are fundamental trigonometric functions that relate the angles and sides of a right triangle. These functions are among the building blocks of trigonometry and appear in many mathematical contexts.

Cosine Function

The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. For angle \( 30^{\circ} \), the cosine value is calculated as:\[\cos 30^{\circ} = \frac{\sqrt{3}}{2}\]

Sine Function

The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. For angle \( 30^{\circ} \), the sine value is:\[\sin 30^{\circ} = \frac{1}{2}\]These functions allow us to understand the relationships within triangles and are used to calculate angles, side lengths, and areas in various geometric problems. Understanding cosine and sine also forms the basis for understanding more complex trigonometric identities.

Angle Measures

Angle measures are critical in both geometry and trigonometry as they define the sharpness or openness of angles within shapes. Angles are most commonly measured in degrees or radians. In trigonometry, you're often working with angle measures to find side lengths, relationships, and to solve trigonometric equations. For instance, in our exercise, we deal with a 30° angle, a common acute angle. Acute angles are those less than 90° and frequently arise in problems involving right triangles.
When evaluating trigonometric expressions, knowing the angle measures such as 0°, 30°, 45°, 60°, and 90° is important, as they often have simple trigonometric values associated with them. Understanding these conceptually will help you not only perform computations but also check if the results make sense given the nature of the angle used.