Find the reference number for each value of t.

Find the terminal point P(x, y) on the unit circle determined by the given value of t. t = 3 pi/2 Find the reference number for each value of t.

(a) t = 13 pi/6
(b) t = -9pi/7
(c) t = 6
(d) t = -7

The Correct Answer and Explanation is:

1. Terminal Point on the Unit Circle

Given:
t=3π2t = \frac{3\pi}{2}

On the unit circle, the coordinates of a point corresponding to an angle tt (in radians) are: P(x,y)=(cos⁡t,sin⁡t)P(x, y) = (\cos t, \sin t)

So: cos⁡(3π2)=0,sin⁡(3π2)=−1\cos\left(\frac{3\pi}{2}\right) = 0,\quad \sin\left(\frac{3\pi}{2}\right) = -1

Answer: P(x,y)=(0,−1)P(x, y) = (0, -1)

2. Reference Number for Each Value of tt

A reference number is the acute angle (positive angle ≤ π2\frac{\pi}{2}) formed by the terminal side of the angle tt and the x-axis. Here’s how to find it for each value:

(a) t=13π6t = \frac{13\pi}{6}

Since this is more than 2π2\pi, subtract 2π2\pi: 13π6−2π=13π−12π6=π6\frac{13\pi}{6} – 2\pi = \frac{13\pi – 12\pi}{6} = \frac{\pi}{6}

Reference number: π6\frac{\pi}{6}

(b) t=−9π7t = -\frac{9\pi}{7}

Add 2π2\pi until you get a positive coterminal angle: −9π7+2π=−9π7+14π7=5π7-\frac{9\pi}{7} + 2\pi = -\frac{9\pi}{7} + \frac{14\pi}{7} = \frac{5\pi}{7}

This is between 0 and π\pi, so the reference angle is just:

Reference number: 5π7\frac{5\pi}{7}

(c) t=6t = 6 (in radians)

Find coterminal angle by subtracting 2π≈6.2832\pi \approx 6.283: 6−2π≈6−6.283=−0.2836 – 2\pi \approx 6 – 6.283 = -0.283

Then add 2π2\pi to make it positive: -0.283 + 2\pi \approx 6.0 \) (original value So reference number is: \[ \pi – 6 \approx 3.142 – 6 = \text{Not valid}

Better: find closest multiple of π\pi, say t≈π−0.283⇒t \approx \pi – 0.283 \Rightarrow reference angle is 2π−6≈0.2832\pi – 6 \approx 0.283

Reference number: ≈2π−6=0.283\approx 2\pi – 6 = 0.283

(d) t=−7t = -7

Add 2π≈6.2832\pi \approx 6.283 multiple times to make it positive: −7+2π≈−7+6.283=−0.717⇒Still negative-7 + 2\pi \approx -7 + 6.283 = -0.717 \Rightarrow \text{Still negative}

Add another 2π2\pi: −0.717+6.283=5.566-0.717 + 6.283 = 5.566

Now find the reference number: 2π−5.566≈6.283−5.566=0.7172\pi – 5.566 \approx 6.283 – 5.566 = 0.717

Reference number: ≈0.717\approx 0.717

Summary of Answers

P(x,y)=(0,−1)P(x, y) = (0, -1)
Reference Numbers:
- (a) π6\frac{\pi}{6}
- (b) 5π7\frac{5\pi}{7}
- (c) ≈0.283\approx 0.283
- (d) ≈0.717\approx 0.717

Explanation

In trigonometry, angles can be measured in radians, and the unit circle helps us understand their behavior. A unit circle has a radius of 1, and any angle tt drawn in standard position will intersect the circle at a point P(x,y)P(x, y), where x=cos⁡tx = \cos t and y=sin⁡ty = \sin t. For example, at t=3π2t = \frac{3\pi}{2}, the terminal side points straight down along the y-axis, so the point on the unit circle is (0,−1)(0, -1).

The reference number is a concept used to simplify angle analysis. It refers to the acute angle that the terminal side of an angle tt makes with the x-axis. To find this number, we often reduce the given angle to an equivalent one between 0 and 2π2\pi by adding or subtracting multiples of 2π2\pi. Once we have the coterminal angle, the reference angle is based on which quadrant it lies in:

Quadrant I: reference angle = tt
Quadrant II: reference angle = π−t\pi – t
Quadrant III: reference angle = t−πt – \pi
Quadrant IV: reference angle = 2π−t2\pi – t

Negative angles rotate clockwise, so we add 2π2\pi repeatedly to get a positive coterminal angle. Non-radian numbers like 6 or -7 are interpreted as radians unless otherwise stated.

By using these techniques, we determined reference angles and the terminal point on the unit circle accurately, reinforcing the importance of unit circle concepts in trigonometry.