Problem 85 - Precalculus

Question

For the following exercises, find the exact value of each trigonometric function. $$ \sin \frac{\pi}{3} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Angle

The problem asks for the value of \( \sin \frac{\pi}{3} \). It’s important to realize that \( \frac{\pi}{3} \) radians is equivalent to 60 degrees.

Step 2: Recall the Sine Value

Recall the sine value for 60 degrees (\( \frac{\pi}{3} \)). From trigonometric tables or the unit circle, we know that \( \sin 60^\circ \) is \( \frac{\sqrt{3}}{2} \).

Step 3: Conclude

Thus, the exact value of \( \sin \frac{\pi}{3} \) is \( \frac{\sqrt{3}}{2} \).

Key Concepts

Sine

Let's explore the sine function, one of the fundamental trigonometric functions that often appears in various math problems. The sine function measures the y-coordinate of a point on the unit circle corresponding to a given angle. In basic terms, for an angle in a right triangle, sine (\(\sin\theta\)) represents the ratio of the length of the opposite side to the hypotenuse:

• Formula: \(\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}\)

In trigonometry, angles can be expressed in degrees or radians. Understanding the sine function in both systems is key to unlocking many math concepts. The sine of 0 degrees or 0 radians is always 0, while the sine of 90 degrees (\(\frac{\pi}{2}\) radians) is 1. These simple instances showcase the intuitive usage of sine.When you look at the unit circle, the sine value signifies how far up or down a point is from the horizontal axis. The sine function is repetitive, changing every 180 degrees or \(\pi\) radians, and forms a smooth wave-like pattern.

Unit Circle

The unit circle is an essential tool for understanding trigonometric functions, including sine. It is a circle with a radius of 1, centered at the origin of the coordinate plane.The unit circle makes it convenient to calculate the values of trigonometric functions because it simplifies their computations using angles in either degrees or radians. Any given point on the unit circle can be expressed as \((\cos \theta, \sin \theta)\) where \(\theta\) is the angle formed with the positive x-axis. These points help visualize how trigonometric functions work.For example:

• At 0 degrees or 0 radians, the point on the unit circle is (1, 0), which shows that \(\sin 0 = 0\).
• At 90 degrees or \(\frac{\pi}{2}\) radians, the point is (0, 1), revealing that \(\sin\frac{\pi}{2} = 1\).
• At 60 degrees or \(\frac{\pi}{3}\) radians, the sine value is \(\frac{\sqrt{3}}{2}\), which can be seen from the coordinate (cos 60°, sin 60°).

The regular pattern of the sine function around the unit circle allows these values to be predictable and repetitive.

Radians to Degrees Conversion

Angles can be measured in radians or degrees, and converting between these two measurements is a crucial skill. While degrees are often used in everyday contexts, radians are more common in mathematical applications, especially trigonometry.To convert from radians to degrees, remember the key relationship:

• \(180\degree = \pi\ ext{ radians}\)

This opens a straightforward method for conversion:

• Degrees = Radians × \(\frac{180}{\pi}\)
• Radians = Degrees × \(\frac{\pi}{180}\)

Using these formulas, you can switch seamlessly between radians and degrees when working with trigonometric functions. For example, to convert \(\frac{\pi}{3}\) radians to degrees:

• Degrees = \(\frac{\pi}{3} \times \frac{180}{\pi} = 60\degree\)

This conversion is especially handy in trigonometry, where angles often begin in radians, while intuitive understanding might be easier in degrees. Having this ability enriches your problem-solving toolkit and maths fluency.