Problem 23 - Algebra and Trigonometry

Question

9–32 Find the exact value of the trigonometric function. $$\sin \frac{3 \pi}{2}$$

Step-by-Step Solution

Answer

\(\sin \frac{3\pi}{2} = -1\)

Step 1: Understand the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It helps us find the sine and cosine values of angles. The coordinates of any point on the unit circle are given by \((\cos \theta, \sin \theta)\), where \(\theta\) is the angle.

Step 2: Locate the Angle on the Unit Circle

The angle \(\frac{3\pi}{2}\) radians corresponds to 270 degrees. On the unit circle, this angle is located at the negative \(y\)-axis (straight down from the circle's center).

Step 3: Determine the Sine Value

At \(270\) degrees or \(\frac{3\pi}{2}\) radians, the coordinate on the unit circle is \((0, -1)\). The sine of an angle in the unit circle is given by the \(y\)-coordinate of the point, which in this case is \(-1\).

Step 4: Write the Exact Value

Using the point \((0, -1)\) on the unit circle, the sine function at \(\frac{3\pi}{2}\) is \(-1\). Therefore, the exact value of \(\sin \frac{3\pi}{2}\) is \(-1\).

Key Concepts

Unit Circle

The unit circle is a fundamental tool in trigonometry. It is a circle that has a radius of one, and its center is located at the origin (0,0) of a coordinate plane. The simplicity of the unit circle makes it easy to find the sine and cosine values of angles, as the x-coordinate represents the cosine, and the y-coordinate represents the sine.

In the unit circle, every angle can be represented as a point, and each point can be expressed in the form. These points make it incredibly straightforward to find exact trigonometric values without needing a calculator. For example:

• The angle of 0° or 0 radians is at the point (1, 0).
• The angle of 90° or \(\frac{\pi}{2}\) radians is at the point (0, 1).
• The angle of 180° or \(\pi\) radians corresponds to the point (-1, 0).
• The angle of 270° or \(\frac{3\pi}{2}\) radians is found at the point (0, -1).

As you can see, these fixed points help simplify the calculation of sine and cosine for key angles.

Sine of an Angle

The sine of an angle is closely related to its position on the unit circle. Specifically, the sine value is the y-coordinate of the point where the terminal side of the angle intersects the unit circle. Understanding sine within this context helps you visualize and remember values easier.

When looking at the unit circle:

• The sine of 0 is 0, because at the angle 0, the y-coordinate is 0.
• The sine of \(\frac{\pi}{2}\) is 1, as the point (0, 1) represents this angle, and the y-coordinate is 1.
• The sine of \(\pi\) is 0, because the coordinate at this angle is (-1, 0).
• The sine of \(\frac{3\pi}{2}\) is -1, where the point (0, -1) denotes the y-value.

These values reflect how the sine behaves as the angle changes, forming a smooth wave-like pattern called the sine wave.

Radian Measure

Radian measure is a way of measuring angles based on the radius of the circle. Unlike degrees, which divide a circle into 360 equal parts, radians use the circle's arc length relative to the radius.

One complete revolution around a circle equals 2\(\pi\) radians. This conversion is essential:

• 90° is equivalent to \(\frac{\pi}{2}\) radians.
• 180° is equivalent to \(\pi\) radians.
• 270° is equivalent to \(\frac{3\pi}{2}\) radians.
• 360° is equivalent to 2\(\pi\) radians.

Radian measure is particularly useful in mathematics because it simplifies many calculations and is the standard unit of measure in most higher mathematics fields. Understanding radian measure allows for a seamless transition from the unit circle to direct calculations and applications in calculus.