Problem 28 - Precalculus Mathematics for Calculus

Question

Evaluate the expression without using a calculator. $$\left(\sin 60^{\circ}\right)^{2}+\left(\cos 60^{\circ}\right)^{2}$$

Step-by-Step Solution

Answer

The expression evaluates to 1, as confirmed by trigonometric identities.

Step 1: Know the Trigonometric Values

We need to know the standard trigonometric values. The value of \( \sin 60^{\circ} \) is \( \frac{\sqrt{3}}{2} \) and the value of \( \cos 60^{\circ} \) is \( \frac{1}{2} \).

Step 2: Square Each Value

Now, square each of the values obtained in Step 1. This means we calculate \((\sin 60^{\circ})^2 = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\) and \((\cos 60^{\circ})^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\).

Step 3: Add the Results

Add the squared values obtained in Step 2. So, \( (\sin 60^{\circ})^2 + (\cos 60^{\circ})^2 = \frac{3}{4} + \frac{1}{4} = 1 \).

Step 4: Verify Using the Pythagorean Identity

According to the Pythagorean identity, for any angle \( \theta \), \( \sin^2 \theta + \cos^2 \theta = 1 \). We've confirmed that this identity holds true for \( 60^{\circ} \), matching our result in Step 3.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the most fundamental relationships in trigonometry. It states that for any angle \( \theta \), the square of the sine of \( \theta \) plus the square of the cosine of \( \theta \) is equal to 1. This can be expressed mathematically as:\[\sin^2 \theta + \cos^2 \theta = 1\]This identity is derived from the Pythagorean theorem itself, applicable in a unit circle, where the hypotenuse is equal to the radius, which is 1. Therefore, any point on the unit circle maintains this identity. In our exercise with an angle of 60 degrees, verifying this identity involves confirming that the sum of \( (\sin 60^{\circ})^2 \) and \( (\cos 60^{\circ})^2 \) equals 1. This consistency underpins much of trigonometric analysis in mathematics.

Standard Trigonometric Values

Trigonometric functions such as sine and cosine are periodic and have certain standard values at common angles. These include 0°, 30°, 45°, 60°, and 90° among others. Knowing these values helps in quick evaluations without a calculator. For 60°, the important values are:

• \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \cos 60^{\circ} = \frac{1}{2} \)

Memorizing these standard values can greatly assist in manual calculations and verifying trigonometric identities. These values are derived from the properties of an equilateral triangle, which divided creates a 30-60-90 triangle, leading directly to these ratios.

Angle Evaluation

When evaluating angles, understanding their positioning and reference angles on the unit circle is crucial. For instance, 60 degrees is located in the first quadrant of the unit circle, where both sine and cosine values are positive. This makes 60° a simple angle for demonstrating basic trigonometric functions. Because trigonometric functions repeat every 360 degrees, knowing values at key benchmarks (like 60°) allows for evaluations of co-terminal angles—those that reach the same point on the circle. Recognizing these positional relationships aids in solving trigonometric problems efficiently.

Manual Calculation

Manual calculations in trigonometry involve methodically applying known formulas and identities without the aid of a calculator. The steps used in the original exercise demonstrate the process:

• Determine standard angle values (e.g., \( \sin 60^{\circ} \), \( \cos 60^{\circ} \)).
• Perform algebraic operations such as squaring these values.
• Add the results to see if they fit known identities like \( \sin^2 \theta + \cos^2 \theta = 1 \).

These steps encourage active engagement with the material and improve conceptual understanding. For a student, each step in a manual calculation builds foundational skills that are applicable across a range of problems within trigonometry.