Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \sin 60^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \(\sin 60^{\circ}\) is \(\frac{\sqrt{3}}{2}\).

Step 1: Understanding the Problem

The problem asks us to find the exact value of \( \sin 60^{\circ} \). The sine function is a trigonometric function that relates an angle of a right triangle to the ratio of the opposite side to the hypotenuse.

Step 2: Using Special Angles

The angle \(60^{\circ}\) is one of the special angles in trigonometry, often found in equilateral or right-angled triangles. These angles usually have known values in terms of radicals.

Step 3: Recalling Exact Values

For the special angle \(60^{\circ}\), the exact value of \(\sin 60^{\circ}\) is \(\frac{\sqrt{3}}{2}\). This value can be derived from the geometry of a 30-60-90 triangle, where the side opposite 60 degrees is \(\sqrt{3/2}\) times the hypotenuse.

Key Concepts

Special Angles

Special angles in trigonometry are specific angles that have unique, exact trigonometric values. These angles include 0°, 30°, 45°, 60°, and 90°.
While they may appear in various geometric scenarios, they are most commonly associated with two specific types of triangles: the equilateral triangle and the isosceles right triangle.
For instance, in a 30-60-90 triangle, the sides are in the ratio of 1:the square root of 3:2. This unique ratio directly helps us find the trigonometric values of 30° and 60°, which are common in many math problems.

• 30°:
  • \(\sin 30^{\circ} = \frac{1}{2}\)
  • \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\)
  • \(\tan 30^{\circ} = \frac{\sqrt{3}}{3}\)
• 60°:
  • \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\)
  • \(\cos 60^{\circ} = \frac{1}{2}\)
  • \(\tan 60^{\circ} = \sqrt{3}\)

By memorizing or understanding the derivation of the values at these angles, solving trigonometric problems becomes easier.

Sine Function

The sine function is one of the fundamental trigonometric functions, abbreviated as \(\sin\). It's a ratio that arises from looking at a right triangle.
In a right triangle, the sine of an angle \(\theta\) is defined as the length of the side opposite to the angle divided by the hypotenuse.
The formula is represented as:\[\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\] Sine values vary as the angle increases or decreases. They range between -1 and 1. Some simple properties and principles related to the sine function include:

• It is periodic with a period of 360° or \(2\pi\) radians.
• It is an odd function, which means \(\sin(-\theta) = -\sin(\theta)\).
• It starts at 0, peaks at 90°, comes back to 0 at 180°, goes negative peaking at -1 at 270°, and closes the cycle back at 360° (or 0°).

These properties reveal how sine behaves, allowing you to predict its values across different quadrants of a unit circle.

Exact Values of Trigonometric Functions

Knowing the exact values of trigonometric functions for special angles is crucial in both basic and advanced mathematical problems. Exact values provide precise answers and are usually more useful than decimal approximations.
For common angles like 30°, 45°, and 60°, triangles with specific side length ratios—derived from geometry—help determine those values.
For example, for a 30-60-90 triangle, the sides are often set as:- Shorter leg = 1- Longer leg = \(\sqrt{3}\)- Hypotenuse = 2Using this triangle setup, you can derive the exact values, such as:

• \(\sin 30^{\circ} = \frac{1}{2}\)
• \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\)

Other triangles, like the 45-45-90 triangle, have sides in the ratio 1:1:\(\sqrt{2}\), allowing you to find:

• \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\)

Practicing these values regularly will make you more fluent in trigonometry without needing a calculator.