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Magazine Chapter 5: Problem 9
Find the exact value of the trigonometric function at the given real number. (a) \(\cos \frac{3 \pi}{4}\) (b) \(\cos \frac{5 \pi}{4}\) (c) \(\cos \frac{7 \pi}{4}\)
Short Answer
Expert verified
(a) \(-\frac{\sqrt{2}}{2}\), (b) \(-\frac{\sqrt{2}}{2}\), (c) \(\frac{\sqrt{2}}{2}\)
Step by step solution
01
Identify the Quadrant
First, determine which quadrant each angle corresponds to by considering their positions on the unit circle. (a) \( \frac{3\pi}{4} \) is in the second quadrant. (b) \( \frac{5\pi}{4} \) is in the third quadrant. (c) \( \frac{7\pi}{4} \) is in the fourth quadrant.
02
Use Reference Angles
Find the reference angle for each given angle, which is the acute angle formed with the x-axis.(a) The reference angle for \( \frac{3\pi}{4} \) is \( \pi - \frac{3\pi}{4} = \frac{\pi}{4} \).(b) The reference angle for \( \frac{5\pi}{4} \) is \( \frac{5\pi}{4} - \pi = \frac{\pi}{4} \).(c) The reference angle for \( \frac{7\pi}{4} \) is \( 2\pi - \frac{7\pi}{4} = \frac{\pi}{4} \).
03
Determine the Sign Based on Quadrant
Determine the sign of the cosine function in each respective quadrant.(a) In the second quadrant, cosine is negative: \( \cos \frac{3\pi}{4} = -\cos \frac{\pi}{4} \).(b) In the third quadrant, cosine is also negative: \( \cos \frac{5\pi}{4} = -\cos \frac{\pi}{4} \).(c) In the fourth quadrant, cosine is positive: \( \cos \frac{7\pi}{4} = \cos \frac{\pi}{4} \).
04
Calculate Cosine of Reference Angles
Calculate the cosine of the reference angle \( \frac{\pi}{4} \). The value is the same for each case due to the reference angle being the same.\( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).
05
Apply Signs to Results
Apply the determined sign from each quadrant to the calculated cosine value of the reference angle.(a) \( \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \).(b) \( \cos \frac{5\pi}{4} = -\frac{\sqrt{2}}{2} \).(c) \( \cos \frac{7\pi}{4} = \frac{\sqrt{2}}{2} \).
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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 really helpful tool for understanding trigonometric functions like cosine, sine, and tangent. It鈥檚 a circle with a radius of 1, centered at the origin of a coordinate system. This makes it easy to relate angles to points on the circle. In the unit circle:
- The x-coordinate of a point is the cosine of the angle.
- The y-coordinate of a point is the sine of the angle.
Reference Angles
A reference angle is like a helper to find the trigonometric function values of any angle. It's the smallest angle that a given angle makes with the x-axis, and it is always an acute angle (less than 90掳 or \( \frac{\pi}{2} \) radians). Reference angles are the same for angles that appear in different quadrants. To find the reference angle:
- For angles in the first quadrant, the reference angle is the angle itself.
- For angles in the second quadrant, subtract the angle from \( \pi \) (or 180掳).
- For angles in the third quadrant, subtract \( \pi \) from the angle.
- For angles in the fourth quadrant, subtract the angle from \( 2\pi \) (or 360掳).
Quadrants
The coordinate plane is divided into four sections called quadrants. These quadrants are numbered from 1 to 4 in a counter-clockwise direction:
- The first quadrant is where both x and y are positive.
- The second quadrant has a positive y and a negative x.
- The third quadrant has both coordinates negative.
- The fourth quadrant has a negative y and a positive x.
- is positive in the first and fourth quadrants.
- is negative in the second and third quadrants.
Cosine Function
The cosine function, often represented as \( \cos \), relates the angle in a right triangle to the ratio of the adjacent side over the hypotenuse. In the unit circle, the cosine of an angle is the x-coordinate of the point where the terminal side of the angle intersects the circle.
- For angles in the unit circle, the cosine function provides the horizontal distance from the origin to the terminal point.
- The function is periodic, repeating every \( 2\pi \) radians or 360掳.
- The cosine function values range between -1 and 1.
