Problem 1 - Contemporary Precalculus

Question

In Exercises \(1-10,\) use the definition (not a calculator) to find the function value. $$\sin (3 \pi / 2)$$

Step-by-Step Solution

Answer

Answer: The value of \(\sin(3\pi/2)\) is \(-1\).

Step 1: Convert the angle to degrees

First, we need to convert the angle \(3\pi/2\) radians to degrees. To do this, we can use the conversion factor \(\frac{180^\circ}{\pi}\). Multiply the angle in radians by the conversion factor. $$3\pi/2 \cdot \frac{180^\circ}{\pi} = \frac{3\cdot 180^\circ}{2} = 270^\circ$$

Step 2: Locate the angle on the unit circle

Now that we have the angle in degrees, locate the angle \(270^\circ\) on the unit circle. The unit circle is divided into four quadrants, each measuring \(90^\circ\). The angle of \(270^\circ\) means we start from the positive x-axis (0 degrees) and rotate counter-clockwise by the given angle which will take us to the negative y-axis. At this point, the coordinates of the point on the unit circle are \((0, -1)\).

Step 3: Determine the sin value from the coordinates

Once we have located the point on the unit circle, we can now determine the sin value of the angle. Recall that for any point on the unit circle \((x, y)\), the sin value corresponds to the y-coordinate. Therefore, the sin value of the angle \(270^\circ\) (or \(3\pi/2\) radians) is: $$\sin(3\pi/2) = -1$$ Thus, the value of \(\sin(3\pi/2)\) is \(-1\).

Key Concepts

Unit Circle

The unit circle is a fundamental tool in trigonometry. It's a circle with a radius of 1 centered at the origin of a coordinate plane.
The unit circle helps in understanding how the values of trigonometric functions change as angles increase or decrease.

• The unit circle is segmented into four quadrants, each representing a 90-degree section of the full 360 degrees.
• The circle's circumference allows angles to be represented in both degrees and radians, providing a visual aid for conversions.
• Points on the unit circle are denoted as \((x, y)\), where \(x\) is the cosine of the angle and \(y\) is the sine.

To find the sine of an angle, we locate the corresponding point on the unit circle, and the y-coordinate of that point is the sine value. For example, at 270 degrees (or \(3\pi/2\) radians), the point is \((0, -1)\), and the sine is \(-1\).

Radians to Degrees Conversion

Radians and degrees are two units for measuring angles.
Radians are based on the radius of a circle, while degrees divide a full circle into 360 parts.

• A full circle in radians is \(2\pi\), equivalent to 360 degrees.
• To convert from radians to degrees, we multiply by \(\frac{180^\circ}{\pi}\).

For the described exercise, we converted \(3\pi/2\) radians into degrees:\[3\pi/2 \cdot \frac{180^\circ}{\pi} = 270^\circ\]This conversion helps situate the angle on the unit circle, facilitating the calculation of trigonometric functions.

Sine Function

The sine function is a primary trigonometric function that shows the ratio of the opposite side to the hypotenuse in a right triangle.
On the unit circle, the sine of an angle is the y-coordinate of the point where the angle’s terminal side intersects the circle.

• The sine function is periodic with a period of \(360^\circ\) (or \(2\pi\) radians), meaning it repeats its values in each cycle.
• Sine values range from -1 to 1, inclusive.
• At specific angles, like 90 degrees \((\frac{\pi}{2}\)), sine is 1; at 270 degrees \((\frac{3\pi}{2}\)), sine is -1.

In the exercise, finding \(\sin(3\pi/2)\) involves identifying the y-coordinate on the unit circle, which is \(-1\). This showcases how sine values correspond directly to positions on the circle.