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Chapter 10: Problem 11

Find the exact value of the cosine and sine of the given angle. $$ \theta=\frac{3 \pi}{2} $$

Short Answer

Expert verified
\( \cos\left(\frac{3\pi}{2}\right) = 0 \) and \( \sin\left(\frac{3\pi}{2}\right) = -1 \).

Step by step solution

01

Identify the Angle

The given angle is \( \theta = \frac{3\pi}{2} \). This is an angle measured in radians, equivalent to 270 degrees. It is positioned along the negative y-axis on the unit circle.
02

Determine the Quadrant

\( \frac{3\pi}{2} \) radians corresponds to an angle that lies on the boundary between the third and fourth quadrants on the unit circle. Specifically, it is equivalent to pointing directly downwards, along the negative y-axis.
03

Use Unit Circle Definition of Sine and Cosine

On the unit circle, any point on the circle can be described by \((\cos(\theta), \sin(\theta))\). For an angle \( \theta = \frac{3\pi}{2} \), the point corresponds to the coordinates \((0, -1)\).
04

Extract Cosine and Sine Values

From the coordinates \((0, -1)\), we identify that \( \cos\left(\frac{3\pi}{2}\right) = 0 \) and \( \sin\left(\frac{3\pi}{2}\right) = -1 \).

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Key Concepts

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

Understanding the Unit Circle
The unit circle is a crucial concept in trigonometry and helps us understand how trigonometric functions like sine and cosine behave. Picture a circle centered at the origin
  • This circle has a radius of one.
  • The equation that represents this circle is \[ x^2 + y^2 = 1.\]
What makes the unit circle special is how it links angles and coordinates. Any point on the circle represents an angle
  • The x-coordinate corresponds to the cosine of the angle.
  • The y-coordinate corresponds to the sine of the angle.
The unit circle is essential for visualizing angle measures and their sine and cosine values, especially when dealing with radians and quadrants.
The Radian Measure
Radians are a way of measuring angles, different from the more familiar degrees. While degrees divide a circle into 360 parts, radians measure angles based on the radius of a circle.
  • One full circle is \(2\pi\) radians, equivalent to 360 degrees.
  • This means that \(\pi\) radians is half a circle, or 180 degrees.
  • For our specific angle, \(\frac{3\pi}{2}\) radians is equivalent to 270 degrees.
Radians are often used in higher mathematics because of their natural relationship with the unit circle. With radians, the formula for arc length becomes simple
  • \(\theta = \frac{s}{r}\)
where \(\theta\) is the angle in radians, \(s\) is the arc length, and \(r\) is the radius.
Identifying Quadrants
The Cartesian plane is divided into four quadrants, each representing a range of angle measures. Each of these quadrants affects the sign of the sine and cosine values. Understanding these can make solving trigonometric problems easier.
  • The first quadrant includes angles from 0 to \(\frac{\pi}{2}\) radians.
  • The second quadrant covers angles from \(\frac{\pi}{2}\) to \(\pi\) radians.
  • The third quadrant ranges from \(\pi\) to \(\frac{3\pi}{2}\) radians.
  • The fourth quadrant includes \(\frac{3\pi}{2}\) to \(2\pi\) radians.
Our specific angle, \(\frac{3\pi}{2}\) radians, lies along the negative y-axis. It marks the boundary line between the third and fourth quadrants. Knowing which quadrant an angle is in helps determine the signs and values for sine and cosine easily.

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Most popular questions from this chapter

Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\) and that the range of arccosecant is \(\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]\) when finding the exact value. \(\operatorname{arccsc}\left(\csc \left(\frac{9 \pi}{8}\right)\right)\)

In Exercises 222 - 233 , find the domain of the given function. Write your answers in interval notation. $$ f(x)=\arcsin (5 x) $$

In Exercises 188 - 207 , solve the equation using the techniques discussed in Example 10.6 .7 then approximate the solutions which lie in the interval \([0,2 \pi)\) to four decimal places. $$ \sin (x)=\frac{7}{11} $$

Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\) and that the range of arccosecant is \(\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]\) when finding the exact value. $$ \operatorname{arcsec}\left(\sec \left(\frac{5 \pi}{6}\right)\right) $$

find the exact value or state that it is undefined. $$ \sin (\arctan (-2)) $$
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