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Chapter 5: Problem 16

Skills This set of exercises will reinforce the skills illustrated in this section. In Exercises \(9-22,\) find the reference angle for each of the angles given. $$-\frac{7 \pi}{4}$$

Short Answer

Expert verified
The reference angle for \(-\frac{7 \pi}{4}\) is \(\frac{\pi}{4}\).

Step by step solution

01

Convert the angle into positive form

First, add \(2\pi\) to the given angle \(-\frac{7 \pi}{4}\) to make it positive. The formula for converting a negative angle into positive is \(angle + 2\pi\). Hence, the positive angle for \(-\frac{7 \pi}{4}\) is \(-\frac{7 \pi}{4} + 2\pi = -\frac{7 \pi}{4} + \frac{8 \pi}{4}= \frac{\pi}{4}\).
02

Identify the reference angle using the unit circle

Once the angle is in positive form, find the reference angle using the unit circle. The reference angle for any angle in standard position is the acute angle formed by the terminal side of that angle and the x-axis. In other words, it's the smallest angle that the terminal side makes with the x-axis. Since the angle obtained in Step 1, \(\frac{\pi}{4}\) lies in the first quadrant and all angles in first quadrant are same as their reference angles, the reference angle for \(\frac{\pi}{4}\) is \(\frac{\pi}{4}\) itself.

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Key Concepts

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

Unit Circle
The unit circle is a circle with a radius of one unit, centered at the origin of a coordinate plane. Understanding the unit circle is essential for studying trigonometry and angles. All points on the unit circle are of the form \((\cos \theta, \sin \theta)\), where \(\theta\) is the angle formed with the positive x-axis.

  • Angles are measured in radians or degrees. In the context of the unit circle, we often use radians.
  • The complete circumference of the unit circle corresponds to an angle of \(2\pi\) radians or \(360^\circ\).
  • The x-axis acts as the starting line for measuring angles on the unit circle, beginning from \(0\) radians.
By understanding where an angle lies on the unit circle, you can easily determine its sine and cosine values, which are critical for finding reference angles and solving trigonometric equations.
Positive Angle Conversion
Converting negative angles to positive ones is a fundamental skill in trigonometry. It's crucial because it allows us to understand angles in a standardized form, which is easier for calculating reference angles.

To convert a negative angle to a positive one:
  • Add \(2\pi\) (or \(360^\circ\)) to the negative angle since these are full rotations on the circle.
  • The resulting angle will now be in the same position on the unit circle as the original, but expressed as a positive measure.
For example, given an angle \(-\frac{7\pi}{4}\), adding \(2\pi\) (which is \(\frac{8\pi}{4}\) in terms of common denominators) gives \(\frac{\pi}{4}\). Using this method helps to identify the smallest positive equivalent of an angle, making trigonometric calculations more straightforward.
Quadrants of Angles
Angles on the unit circle are divided into four quadrants, each representing a range of angle measures. Understanding which quadrant an angle resides in helps in finding its reference angle and applying trigonometric identities.

  • First Quadrant: Angles between \(0\) and \(\frac{\pi}{2}\) radians (or \(0\) to \(90^\circ\)). Here, all trigonometric functions are positive.
  • Second Quadrant: Angles between \(\frac{\pi}{2}\) and \(\pi\) radians (or \(90^\circ\) to \(180^\circ\)). Sine is positive, while cosine and tangent are negative.
  • Third Quadrant: Angles between \(\pi\) and \(\frac{3\pi}{2}\) radians (or \(180^\circ\) to \(270^\circ\)). Here, tangent is positive, but sine and cosine are negative.
  • Fourth Quadrant: Angles between \(\frac{3\pi}{2}\) and \(2\pi\) radians (or \(270^\circ\) to \(360^\circ\)). Cosine is positive, whereas sine and tangent are negative.
By determining the quadrant of an angle, you can further deduce the sign of its trigonometric values, aiding in accurate finding of reference angles and solutions.

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Most popular questions from this chapter

Graph at least two cycles of the given functions. $$f(x)=-2 \sin \left(x-\frac{\pi}{4}\right)$$

In this set of exercises, you will use degree and radian measure to study real-world problems. In the 1800 s, women often carried pleated fans. One of the fans on display at the Smithsonian is 7 inches long and, when fully open, sweeps out an angle of \(80^{\circ}\) How long is the trim, to the nearest tenth of an inch, on the curved edge of the fan?

Graph the given pair of functions in the same window. Graph at least two cycles of each function, and describe the similarities and differences between the graphs. $$f(x)=\cot (3 \pi x) ; f(x)=\cot \left(\frac{\pi}{3} x\right)$$

Find the exact values of the given expressions in radian measure. $$\cot ^{-1}(-1)$$

For Exercises \(61-72,\) fill in the given table with the missing information. A pproximate all nonexact answers to four decimal places. $$ \begin{array}{|r|c|c|c|c|} \hline & \text { Quadrant } & \sin t & \cos t & \tan t \\ \hline 61 . & \mathrm{I} & \frac{1}{2} & & \\ \hline 62 . & \mathrm{IV} & & \frac{1}{2} & \\ \hline 63 . & \mathrm{III} & & & 1 \\ \hline 64 . & \mathrm{II} & & & -1 \\ \hline 65 . & \mathrm{II} & & -\frac{1}{2} & \\ \hline 66 . & \mathrm{II} & & -\frac{\sqrt{3}}{2} & \\ \hline 67 . & \mathrm{IV} & -0.6 & & \\ \hline 68 . & \mathrm{III} & -0.8 & & \\ \hline 69 . & \mathrm{II} & & -\frac{5}{13} & \\ \hline 70 . & \mathrm{IV} & & \frac{12}{13} & \\ \hline 71 . & \mathrm{IV} & & & -2 \\ \hline 72 . & \mathrm{II} & & & \\ \hline \end{array} $$
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