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Chapter 4: Problem 58

Evaluate the following expressions exactly: $$\cos 120^{\circ}$$

Short Answer

Expert verified
The exact value is \(-\frac{1}{2}\).

Step by step solution

01

Understand the Cosine Angle

We need to evaluate \( \cos 120^{\circ} \). Notice that \(120^{\circ}\) is an angle in the second quadrant of the unit circle.
02

Find Reference Angle

The reference angle for \(120^{\circ}\) is found by subtracting from \(180^{\circ}\) (since \(120^{\circ}\) is in the second quadrant). Thus, the reference angle is \( 180^{\circ} - 120^{\circ} = 60^{\circ} \).
03

Use the Cosine Function

In the second quadrant, the cosine of an angle is negative, and the cosine of the reference angle \(60^{\circ}\) is \(\frac{1}{2}\). Therefore, \( \cos 120^{\circ} = -\frac{1}{2} \).

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Key Concepts

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

Cosine Function
The cosine function is a fundamental trigonometric function that reflects the horizontal coordinate of an angle on the unit circle. It measures how far along the x-axis the point corresponding to an angle is, as you move counterclockwise from the positive x-axis.
  • The cosine of an angle in a right triangle is the ratio of the adjacent side over the hypotenuse.
  • The cosine function is periodic with a cycle of 360掳 (or 2蟺 radians), meaning it repeats every full rotation of the circle.
  • In the unit circle, as the angle increases from 0掳 to 360掳, the cosine value starts at 1, decreases to -1, and returns to 1.
When working with angles beyond the first quadrant (0掳 to 90掳), it's important to remember that the cosine value can be both positive and negative depending on the quadrant:
  • First Quadrant (0掳 to 90掳): Cosine values are positive.
  • Second Quadrant (90掳 to 180掳): Cosine values are negative.
  • Third Quadrant (180掳 to 270掳): Cosine values remain negative.
  • Fourth Quadrant (270掳 to 360掳): Cosine values return to positive.
Unit Circle
The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It is a powerful tool in trigonometry to understand the relationship between cosine and other trigonometric functions. Here's why the unit circle is essential:
  • Standard Reference: The unit circle provides a standard way to express cosine (and sine) values for different angles.
  • Quadrants and Signs: As a circle is divided into four quadrants, it helps determine whether the trigonometric functions are positive or negative.
  • Angles in Radians and Degrees: The unit circle allows conversion between angle measures: degrees and radians. For example, 360掳 corresponds to 2蟺 radians.
By positioning the terminal side of angles along the circumference of the unit circle, you can easily calculate and memorize cosine values for angles:
  • 0掳 and 360掳 correspond to a cosine value of 1.
  • 180掳 gives a cosine of -1.
  • 90掳 and 270掳 yield a cosine of 0.
Reference Angle
A reference angle is a tool to simplify finding trigonometric function values for any given angle. It is the smallest angle a given angle makes with the x-axis. For angles greater than 90掳, reference angles keep calculations manageable:
  • For the first quadrant, the reference angle is the angle itself.
  • In the second quadrant, subtract the angle from 180掳. For example, for 120掳, the calculation is: 180掳 - 120掳 = 60掳.
  • For the third quadrant, subtract 180掳 from the angle.
  • In the fourth quadrant, subtract the angle from 360掳.
Understanding the concept of reference angles makes it easier to find the cosine (and other trigonometric function values) for non-standard angles:
  • Reference angles allow you to use a known base angle to find trigonometric values in another quadrant.
  • This is particularly useful in conjunction with the signs determined by the unit circle quadrants for accurate calculations.

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Most popular questions from this chapter

Evaluate the expression \(\sec 120^{\circ}\) exactly. Solution: \(120^{\circ}\) lies in quadrant 11 The reference angle is \(30^{\circ}\) \(\begin{array}{l}\text { Find the cosine of the } \\ \text { reference angle. }\end{array} \quad \cos 30^{\circ}=\frac{\sqrt{3}}{2}\) \(\begin{array}{l}\text { Cosine is negative in } \\ \text { quadrant II. }\end{array} \quad \cos 120^{\circ}=-\frac{\sqrt{3}}{2}\) \(\begin{array}{l}\text { Secant is the reciprocal } \\ \text { of cosine. }\end{array} \quad \sec 120^{\circ}=-\frac{2}{\sqrt{3}}=-\frac{2 \sqrt{3}}{3}\) This is incorrect. What mistake was made? (GRAPH CANNOT COPY)

If \(\csc \theta=-\frac{a}{b},\) where \(a\) and \(b\) are positive, and if \(\theta\) lies in quadrant IV, find cot \(\theta\)

Use a calculator to evaluate the following expressions. If you get an error, explain why. $$\tan 270^{\circ}$$

Refer to the following: NASA explores artificial gravity as a way to counter the physiologic effects of extended weightlessness for future space exploration. NASA's centrifuge has a 58 -foot-diameter arm. If two humans are on opposite (red and blue) ends of the centrifuge and their linear speed is 200 miles per hour, how fast is the arm rotating? Express the answer in radians per second to two significant digits.

Find the smallest positive measure of \(\theta\) (rounded to the nearest degree) if the indicated information is true. \(\sec \theta=1.0001\) and the terminal side of \(\theta\) lies in quadrant I.
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