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Chapter 5: Problem 48

Find (a) the reference number for each value of \(t\) and (b) the terminal point determined by \(t\). $$t=\frac{31 \pi}{6}$$

Short Answer

Expert verified
Reference angle: \(30^\circ\). Terminal point: \((-\frac{\sqrt{3}}{2}, -\frac{1}{2})\).

Step by step solution

01

Convert to Degrees

Start by converting the given radian measure \( t = \frac{31\pi}{6} \) into degrees. Use the conversion formula \( degrees = radians \times \frac{180}{\pi} \). \[ t = \frac{31\pi}{6} \times \frac{180}{\pi} = \frac{31 \times 180}{6} = 930 \text{ degrees} \]
02

Determine Coterminal Angle

Find the smallest positive coterminal angle by subtracting \(360\) degrees until the angle is in the 0 to 360-degree range. \[ 930 - 360 \times 2 = 930 - 720 = 210 \text{ degrees} \] Therefore, \( 210 \text{ degrees} \) is coterminal with \( 930 \text{ degrees} \).
03

Find the Reference Angle

The reference angle is the acute angle formed with the x-axis. Since \(210\) degrees is in the third quadrant, subtract \(180\) from it. \[ 210 - 180 = 30 \text{ degrees} \] So, the reference angle is \(30\text{ degrees}\).
04

Determine Terminal Point

Use the reference angle to determine the terminal point on the unit circle for \( t = \frac{31\pi}{6} \). In the third quadrant, the cosine and sine are both negative. Thus, the terminal point is given by \( (-\cos(30^\circ), -\sin(30^\circ)) = \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right) \).

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Key Concepts

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

Reference Angle
A reference angle is the smallest angle that a given angle makes with the x-axis. It helps to simplify trigonometric calculations and always falls between 0掳 and 90掳. In the context of the exercise, the reference angle is found after identifying which quadrant the angle is in.

We determine the reference angle by considering the angle's position:
  • If the angle is in the first quadrant (0掳 to 90掳), the angle itself is the reference angle.
  • If it's in the second quadrant (90掳 to 180掳), subtract the angle from 180掳.
  • In the third quadrant (180掳 to 270掳), as seen in our problem where 210掳 is in third quadrant, subtract 180掳 from the angle.
  • If it's in the fourth quadrant (270掳 to 360掳), subtract the angle from 360掳.
For the problem given, we subtracted 180掳 from 210掳 to get a reference angle of 30掳. This simplifies our task of using the trigonometric functions, as the trigonometric values of 30掳 are well-known and straightforward to work with.
Coterminal Angles
Coterminal angles are angles that share the same terminal side, meaning they differ by full circles. If you keep adding or subtracting 360掳 (a full circle) to an angle, the resulting angles are coterminal.

For example, angles like 30掳, 390掳 (30掳 + 360掳), and -330掳 (30掳 - 360掳) are all coterminal because they end at the same point on a circle. This concept is useful to find an equivalent angle within the standard 0掳 to 360掳 range for any angle.

In our exercise, we started with an angle of 930掳 after conversion from radians. To find a positive coterminal angle, we subtracted 360掳 twice (720掳 in total) and ended up with 210掳. This calculation shows how manipulating angles in terms of coterminality allows us to simplify angles to a standard form that is often easier to analyze.
Unit Circle
The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It serves as a fundamental tool in trigonometry to determine the values of trigonometric functions for any angle. Every angle corresponds to a specific point on this circle, allowing direct computation of sine and cosine values.

Think of the unit circle as a way to visualize angles beyond just their degrees or radians. For instance, angles of 0掳, 90掳, 180掳, and 270掳 hit significant points on the circle, but smaller divisions like 30掳 have known coordinates too.
  • At an angle of 30掳, the coordinates are (\( rac{ ext{鈭�3}}{2}\), \( rac{1}{2}\)). In the third quadrant, both these values change signs due to the negative values of sine and cosine, transforming them to (-\( rac{ ext{鈭�3}}{2}\), -\( rac{1}{2}\)).With this insight, terminal points for our \(t = rac{31 ext{蟺}}{6}\) are easily found using the reference angle of 30掳 on the unit circle.
Understanding the unit circle is crucial since it connects angle measures with geometric interpretations, facilitating computation and strengthening comprehension of trigonometric concepts.

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Most popular questions from this chapter

Find the maximum and minimum values of the function. $$y=x-2 \sin x, 0 \leq x \leq 2 \pi$$

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$y=\cos \left(\frac{\pi}{2}-x\right)$$

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\) $$\left(-\frac{6}{7}, \frac{\sqrt{13}}{7}\right)$$

Find the value of each of the six trigonometric functions (if it is defined) at the given real number \(t\). Use your answers to complete the table. $$\begin{array}{|c|c|c|c|c|c|c|} \hline t & \sin t & \cos t & \tan t & \csc t & \sec t & \cot t \\ \hline 0 & 0 & 1 & & \text { undefined } & & \\ \hline \frac{\pi}{2} & & & & & & \\ \hline \pi & & & 0 & & & \text { undefined } \\ \hline \frac{3 \pi}{2} & & & & & & \\ \hline \end{array}$$ $$t=\pi$$

Find the period and graph the function. $$y=\sec \left(3 x+\frac{\pi}{2}\right)$$
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