Problem 67 - Precalculus: A Concise Course

Question

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$ -\frac{3 \pi}{2} $$

Step-by-Step Solution

Answer

\(\sin(-\frac{3 \pi}{2}) = -1\), \(\cos(-\frac{3 \pi}{2}) = 0\), \(\tan(-\frac{3 \pi}{2})\) is undefined.

Step 1: Identification

Given an angle \(-\frac{3 \pi}{2}\). Note that this angle lies in the negative direction of the y-axis when drawn on the unit circle.

Step 2: Calculation of the Sine

The sine of an angle in the unit circle is the y-coordinate of the point where the terminal side of the angle intersects the circle. Therefore, for the angle \(-\frac{3 \pi}{2}\), since it lies on the negative y-axis, the y-coordinate of this point is -1. Hence, \(\sin(-\frac{3 \pi}{2}) = -1\)

Step 3: Calculation of the Cosine

The cosine of an angle in the unit circle is the x-coordinate of the point where the terminal side of the angle intersects the circle. Since the angle \(-\frac{3 \pi}{2}\) lies on the y-axis, its x-coordinate is 0. Hence, \(\cos(-\frac{3 \pi}{2}) = 0\)

Step 4: Calculation of the Tangent

The tangent of an angle is the ratio of the sine to the cosine of that angle. Therefore, the tangent of the angle \(-\frac{3 \pi}{2}\) is given by \(\tan(-\frac{3 \pi}{2}) = \frac{\sin(-\frac{3 \pi}{2})}{\cos(-\frac{3 \pi}{2})}\). Plugging in the derived values of sine and cosine, we get \(\tan(-\frac{3 \pi}{2}) = \frac{-1}{0}\). But the division with 0 is undefined, hence, \(\tan(-\frac{3 \pi}{2})\) is undefined.

Key Concepts

Sine

The sine function is a fundamental trigonometric function. It helps us understand the vertical position of an angle when it is drawn in a unit circle. The unit circle is a circle with a radius of 1. The sine of an angle is represented by the y-coordinate of the intersection point of the angle's terminal side and the unit circle.
For example, consider an angle of \(-\frac{3 \pi}{2}\). Draw this angle in the unit circle starting from the positive x-axis and turning in the clockwise direction.

• The terminal side of \(-\frac{3 \pi}{2}\) ends at the negative y-axis because the angle completes one full turn and goes another quarter turn.
• The y-coordinate at this position is -1.
• Thus, the sine of \(-\frac{3 \pi}{2}\) is \(\sin(-\frac{3 \pi}{2}) = -1\).

Cosine

Cosine is another fundamental trigonometric function, specifically related to horizontal positions. This function gives us the x-coordinate of the point where an angle's terminal side intersects the unit circle. Cosine complements sine as they provide a full picture of an angle's position.For the angle \(-\frac{3 \pi}{2}\), the process is straightforward:

• As you draw the angle in the unit circle, the terminal side lands on the negative y-axis.
• On the unit circle, the x-coordinate of the negative y-axis is 0.
• Therefore, \(\cos(-\frac{3 \pi}{2}) = 0\).

Tangent

Tangent is a trigonometric function that links sine and cosine. It represents the slope of the angle, calculated as the ratio of the sine to the cosine of the angle. Hence, it provides a relationship between vertical and horizontal movements within the circle.For our given angle \(-\frac{3 \pi}{2}\):

• The sine value is -1, as seen in previous steps.
• The cosine value is 0.
• To find the tangent: \( \tan(-\frac{3 \pi}{2}) = \frac{-1}{0}\).

Remember, division by zero is undefined in mathematics. Thus, the tangent of \(-\frac{3 \pi}{2}\) is considered undefined.

Unit Circle

The unit circle is a crucial tool in trigonometry. It provides a visual representation of angles and helps in understanding the sine, cosine, and tangent functions. This circle has a radius of 1, centered at the origin (0,0) in a coordinate plane.Let's break down its importance:

• Each angle forms a point on the unit circle.
• The x-coordinate of this point gives the cosine, while the y-coordinate gives the sine.
• For angles on axes, such as \(-\frac{3 \pi}{2}\), visualizing the terminal side makes interpretation straightforward.

The unit circle standardizes how we measure and comprehend trigonometric functions, ensuring they are consistent and precise across different angles.