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Chapter 4: Problem 67

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

Short Answer

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

Step by step solution

01

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.
02

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, \(\sin(-\frac{3 \pi}{2}) = -1\)
03

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, \(\cos(-\frac{3 \pi}{2}) = 0\)
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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