Problem 27 - Precalculus Mathematics for Calculus

Question

Evaluate the expression without using a calculator. $$\sin 30^{\circ} \cos 60^{\circ}+\sin 60^{\circ} \cos 30^{\circ}$$

Step-by-Step Solution

Answer

The expression evaluates to 1.

Step 1: Identify Known Values

To evaluate the expression, identify the values of the sine and cosine functions for standard angles. Recall that \( \sin 30^{\circ} = \frac{1}{2} \), \( \cos 60^{\circ} = \frac{1}{2} \), \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \), and \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \).

Step 2: Substitute Values into the Expression

Replace each trigonometric function in the expression with its corresponding value. The expression becomes: \[ \left( \frac{1}{2} \right) \left( \frac{1}{2} \right) + \left( \frac{\sqrt{3}}{2} \right) \left( \frac{\sqrt{3}}{2} \right) \].

Step 3: Simplify Each Term

Calculate each term in the expression: \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \) and \( \frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2} = \frac{3}{4} \).

Step 4: Add the Terms Together

Add the two results obtained from the simplification: \( \frac{1}{4} + \frac{3}{4} = 1 \).

Key Concepts

Standard Angles

In trigonometry, **standard angles** refer to specific angle measures often used because of their simple trigonometric values. These angles are typically found on the unit circle and include common angles such as \(30^{\circ}\), \(45^{\circ}\), and \(60^{\circ}\), among others. These angles are significant because they have precise sine and cosine values that can be easily memorized and applied to solve problems without requiring a calculator.

For example, knowing that \(\sin 30^{\circ} = \frac{1}{2}\) and \(\cos 60^{\circ} = \frac{1}{2}\) allows for quick calculations in trigonometric expressions. The same applies to \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\) and \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\). These values form the backbone of solving trigonometric problems involving standard angles, since they are consistent and do not change.

When dealing with solutions that require precise results, such as evaluating trigonometric expressions, these standard angles provide an efficient way to determine the outcome without computation errors.

Sine Function

The **sine function** is one of the fundamental trigonometric functions used to relate the angle of a right triangle to the ratios of its sides. In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. Thus, for an angle \( \theta \), \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).

This function is also graphically represented as a wave that repeats every \(360^{\circ}\) or \(2\pi\) radians, known as its period. It starts at 0, rises to 1, returns to 0, decreases to -1, and then returns to 0. This wave-like pattern is fundamental to waveform analysis in various fields such as physics and engineering.

Applying the sine function to standard angles, we retrieve specific known values that simplify solving trigonometric expressions. For example, \(\sin 30^{\circ} = \frac{1}{2}\) is a directly retrievable value used in calculation exercises, like in the original provided problem, enabling straightforward calculation without a calculator.

Cosine Function

The **cosine function**, closely related to the sine function, represents another essential trigonometric function. In a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse, formulated as \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \).

Similar to the sine function, cosine produces a wave pattern, this time starting at its maximum value of 1. Just like sine, it also has a period of \(360^{\circ}\) or \(2\pi\) radians. These periodic properties make cosine a critical function in modeling periodic phenomena such as circular motion and waves.

For standard angles, knowing values like \(\cos 60^{\circ} = \frac{1}{2}\) and \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\) provides a shortcut to solving trigonometric equations swiftly. These standard values allow students to quickly substitute them into expressions to evaluate without lengthy calculations. This approach was used effectively to solve the original exercise by inserting these well-known cosine values into the expression, leading to a rapid and accurate solution.