Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 11
For each of the following angles, find the reference angle and which quadrant the angle lies in. Then compute sine and cosine of the angle. a. \(\frac{5 \pi}{4}\) b. \(\frac{7 \pi}{6}\) c. \(\frac{5 \pi}{3}\) d. \(\frac{3 \pi}{4}\)
Short Answer
Expert verified
a. 3rd quadrant, \(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\); b. 3rd quadrant, \(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\); c. 4th quadrant, \(-\frac{\sqrt{3}}{2}, \frac{1}{2}\); d. 2nd quadrant, \(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\).
Step by step solution
01
Determine the Quadrant for \(\frac{5 \pi}{4}\)
The angle \(\frac{5 \pi}{4}\) radians is equivalent to \(\frac{5 \times 180}{4} = 225^\circ\). Since 225掳 is between 180掳 and 270掳, the angle lies in the third quadrant.
02
Find the Reference Angle for \(\frac{5 \pi}{4}\)
In the third quadrant, the reference angle is the angle to the closest x-axis. Compute: 225掳 - 180掳 = 45掳. Therefore, the reference angle is \(\frac{\pi}{4}\).
03
Calculate Sine and Cosine for \(\frac{5 \pi}{4}\)
In the third quadrant, both sine and cosine values are negative. Using the reference angle \(\frac{\pi}{4}\), we have: \(\sin\left(\frac{5\pi}{4}\right)=-\frac{\sqrt{2}}{2}\) and \(\cos\left(\frac{5\pi}{4}\right)=-\frac{\sqrt{2}}{2}\).
04
Determine the Quadrant for \(\frac{7 \pi}{6}\)
The angle \(\frac{7 \pi}{6}\) radians is equivalent to \(\frac{7 \times 180}{6} = 210^\circ\). Since 210掳 is between 180掳 and 270掳, the angle lies in the third quadrant.
05
Find the Reference Angle for \(\frac{7 \pi}{6}\)
In the third quadrant, the reference angle is the angle to the closest x-axis. Compute: 210掳 - 180掳 = 30掳. Therefore, the reference angle is \(\frac{\pi}{6}\).
06
Calculate Sine and Cosine for \(\frac{7 \pi}{6}\)
In the third quadrant, both sine and cosine values are negative. Using the reference angle \(\frac{\pi}{6}\), we have: \(\sin\left(\frac{7\pi}{6}\right)=-\frac{1}{2}\) and \(\cos\left(\frac{7\pi}{6}\right)=-\frac{\sqrt{3}}{2}\).
07
Determine the Quadrant for \(\frac{5 \pi}{3}\)
The angle \(\frac{5 \pi}{3}\) radians is equivalent to \(\frac{5 \times 180}{3} = 300^\circ\). Since 300掳 is between 270掳 and 360掳, the angle lies in the fourth quadrant.
08
Find the Reference Angle for \(\frac{5 \pi}{3}\)
In the fourth quadrant, the reference angle is the angle to the closest x-axis. Compute: 360掳 - 300掳 = 60掳. Therefore, the reference angle is \(\frac{\pi}{3}\).
09
Calculate Sine and Cosine for \(\frac{5 \pi}{3}\)
In the fourth quadrant, the sine value is negative, and the cosine value is positive. Using the reference angle \(\frac{\pi}{3}\), we have: \(\sin\left(\frac{5\pi}{3}\right)=-\frac{\sqrt{3}}{2}\) and \(\cos\left(\frac{5\pi}{3}\right)=\frac{1}{2}\).
10
Determine the Quadrant for \(\frac{3 \pi}{4}\)
The angle \(\frac{3 \pi}{4}\) radians is equivalent to \(\frac{3 \times 180}{4} = 135^\circ\). Since 135掳 is between 90掳 and 180掳, the angle lies in the second quadrant.
11
Find the Reference Angle for \(\frac{3 \pi}{4}\)
In the second quadrant, the reference angle is the angle to the closest x-axis. Compute: 180掳 - 135掳 = 45掳. Therefore, the reference angle is \(\frac{\pi}{4}\).
12
Calculate Sine and Cosine for \(\frac{3 \pi}{4}\)
In the second quadrant, the sine value is positive, and the cosine value is negative. Using the reference angle \(\frac{\pi}{4}\), we have: \(\sin\left(\frac{3\pi}{4}\right)=\frac{\sqrt{2}}{2}\) and \(\cos\left(\frac{3\pi}{4}\right)=-\frac{\sqrt{2}}{2}\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sine
The sine function is one of the primary trigonometric functions. It measures the y-coordinate of the point on the unit circle that corresponds to an angle, often used in relation to right triangles.
- In a right triangle, the sine of an angle is the length of the side opposite the angle divided by the hypotenuse.
- This function is periodic and follows a smooth waveform shape, repeating every 360掳 (or 2\(\pi\) radians).
- Sine values are positive in the first and second quadrants.
- Sine values are negative in the third and fourth quadrants.
Cosine
Cosine is another important trigonometric function, similar to sine but focused on the x-coordinate of the point on the unit circle.
- In a right triangle, the cosine of an angle is the length of the side adjacent to the angle divided by the hypotenuse.
- Cosine, like sine, is also periodic and repeats every 360掳 (2\(\pi\) radians).
- Cosine values are positive in the first and fourth quadrants.
- Cosine values are negative in the second and third quadrants.
Reference Angle
The reference angle is a fundamental concept that simplifies finding the sine and cosine of an angle.
- A reference angle is always between 0掳 and 90掳 (or 0 and \(\pi/2\) radians).
- It is the acute angle formed by the terminal side of the given angle and the horizontal axis.
- In the first quadrant, the reference angle is the angle itself.
- In the second quadrant, subtract the angle from 180掳 (or \(\pi\) radians).
- In the third quadrant, subtract 180掳 (or \(\pi\) radians) from the angle.
- In the fourth quadrant, subtract the angle from 360掳 (or 2\(\pi\) radians).
Quadrants
The unit circle, which helps visualize trigonometric functions, is divided into four quadrants.
- First Quadrant: Angles from 0掳 to 90掳 (0 to \(\pi/2\)): Both sine and cosine are positive.
- Second Quadrant: Angles from 90掳 to 180掳 (\(\pi/2\) to \(\pi\)): Sine is positive, cosine is negative.
- Third Quadrant: Angles from 180掳 to 270掳 (\(\pi\) to 3\(\pi/2\)): Both sine and cosine are negative.
- Fourth Quadrant: Angles from 270掳 to 360掳 (3\(\pi/2\) to 2\(\pi\)): Sine is negative, cosine is positive.
