Problem 30 - Algebra 2

Question

Find the exact value of each function. $$ \left(\sin 30^{\circ}\right)^{2}+\left(\cos 30^{\circ}\right)^{2} $$

Step-by-Step Solution

Answer

The exact value is 1.

Step 1: Recall the Trigonometric Identity

Use the Pythagorean identity for sine and cosine: \( \sin^2 \theta + \cos^2 \theta = 1 \) for any angle \( \theta \).

Step 2: Substitute the Given Angle

In the given problem, \( \theta = 30^{\circ} \). Substitute \( \theta = 30^{\circ} \) into the identity: \( \sin^2 30^{\circ} + \cos^2 30^{\circ} = 1 \).

Step 3: Calculate Each Trigonometric Function

For confirmation, calculate \( \sin 30^{\circ} = \frac{1}{2} \) and \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \).

Step 4: Substitute Values into the Expression

Substitute the values into the expression: \( \left(\frac{1}{2}\right)^{2} + \left(\frac{\sqrt{3}}{2}\right)^{2} = \frac{1}{4} + \frac{3}{4} \).

Step 5: Simplify the Expression

Add up the fractions from Step 4: \( \frac{1}{4} + \frac{3}{4} = 1 \).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry. It is one of the key relationships among the basic trigonometric functions: sine and cosine. The identity is expressed as: \( \sin^2 \theta + \cos^2 \theta = 1 \) where \( \theta \) represents an angle. This equation states that for any angle, the sum of the squares of sine and cosine equals one. This identity is derived from the Pythagorean theorem and reflects the unit circle framework, where the hypotenuse is always 1.

• For example, if you set \( \theta = 30^{\circ} \), you can plug this angle into the identity and it holds true: \( \sin^2 30^{\circ} + \cos^2 30^{\circ} = 1 \).
• This identity is valid for all angles, making it a powerful tool in solving trigonometric problems and verifying calculations.

Sine and Cosine Values

Understanding the values of sine and cosine at various angles is essential in trigonometry. For the angle \( 30^{\circ} \), the standard values are:

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

These values come from the ratios of the sides of a 30-60-90 triangle. In such a triangle, the side opposite the 30-degree angle is half the hypotenuse, and the side adjacent is half times the square root of three. These specific ratios are frequently used in various calculations in trigonometry.
If you encounter an angle, knowing these values lets you substitute immediately, simplifying the problem-solving process and ensuring accuracy.

Angle Substitution

Angle substitution involves replacing the variable angle \( \theta \) with a specific angle while ensuring mathematical identities and functions still hold true. In our case, we substitute \( \theta \) with \( 30^{\circ} \) in the Pythagorean identity:

• If \( \theta = 30^{\circ} \), then our identity becomes \( \sin^2 30^{\circ} + \cos^2 30^{\circ} = 1 \).
• This approach allows the use of known, specific values for sine and cosine to verify or simplify equations, ensuring reliable outcomes.

Ultimately, angle substitution is a straightforward way to tailor general trigonometric principles to particular cases, making it easier to compute exact values and solve problems without extensive recalculations or complex conversions.