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Chapter 5: Problem 15

Sketch the angles in standard position. $$\frac{7 \pi}{4}$$

Short Answer

Expert verified
The angle \(\frac{7 \pi}{4}\) radians in standard position will lie in the fourth quadrant of the unit circle, 45掳 clockwise from the positive x-axis.

Step by step solution

01

Understanding the Concept

The notation \(\frac{7 \pi}{4}\) indicates an angle measured in radians. Since \(2\pi\) radians amount to one full circle (360 degrees), \( \frac{\pi}{4}\) can be understood to be 45 degrees. Thus, \( \frac{7 \pi}{4}\) equals \(7 \times 45\) degrees or 315 degrees.
02

Sketching the Unit Circle

Draw a circle with the origin '(0,0)' at its center. Set up the x and y axes, with positive directions to the right and upward and negative directions to the left and downward respectively.
03

Marking the Angle

Starting from the positive x-axis (the initial side), mark an angle of 315 degrees or \(\frac{7 \pi}{4}\) radians in the counterclockwise direction. The terminal side of the angle will lie in the fourth quadrant of the unit circle.

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Key Concepts

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

Radian Measure
Radians are a unit of angular measurement. Unlike degrees, radians provide a more natural way of expressing angles, particularly in mathematics. One full circle is equivalent to \(2\pi\) radians, just like it is equivalent to 360 degrees. This means that the relationship between degrees and radians is straightforward:
  • 180 degrees = \(\pi\) radians
  • 90 degrees = \(\frac{\pi}{2}\) radians
  • A smaller segment, like \(\frac{\pi}{4}\), equals 45 degrees
To convert from degrees to radians, use the formula: \(\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\). For example, \(\frac{7\pi}{4}\) radians corresponds to \(7 \times 45 = 315\) degrees.
Unit Circle
The unit circle is a fundamental concept in trigonometry. It is a circle with its center at the origin of a coordinate plane and a radius of 1 unit. The unit circle allows us to define trigonometric functions for all angles:
  • The x-coordinate of a point on the unit circle represents the cosine of the angle.
  • The y-coordinate represents the sine of the angle.
This standardized circle helps in understanding how angles correspond to points on a plane.
As you measure angles around the circle from the positive x-axis, the movement helps visualize angles in radians and degrees.
Standard Position Angles
Angles in standard position have their vertex at the origin and their initial side on the positive x-axis. This positioning offers a reliable way to visualize and measure angles based on their direction and magnitude.
An angle can be measured:
  • In a counterclockwise direction, which is positive.
  • Clockwise direction gives a negative measurement.
For example, the angle \(\frac{7\pi}{4}\) radians is positioned starting from the positive x-axis and moving counterclockwise, landing in the fourth quadrant of the unit circle.
Quadrants of the Unit Circle
The unit circle is divided into four areas known as quadrants. Each quadrant helps specify the signs of sine and cosine values for angles.
  • First Quadrant - angles from 0 to \(\frac{\pi}{2}\) radians, both sine and cosine are positive.
  • Second Quadrant - angles from \(\frac{\pi}{2}\) to \(\pi\) radians, sine is positive, cosine is negative.
  • Third Quadrant - angles from \(\pi\) to \(\frac{3\pi}{2}\) radians, both sine and cosine are negative.
  • Fourth Quadrant - angles from \(\frac{3\pi}{2}\) to \(2\pi\) radians, sine is negative and cosine is positive.
Understanding these quadrants helps locate the terminal side of any angle, such as \(\frac{7\pi}{4}\) radians, which resides in the fourth quadrant.

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Most popular questions from this chapter

Graph at least two cycles of the given functions. $$f(x)=3 \cos \left(x+\frac{\pi}{2}\right)$$

Suppose \(t\) is in \(\left(0, \frac{\pi}{2}\right) .\) Express \(\sin \left(t+\frac{\pi}{2}\right)\) in terms of sin \(t .\) (Hint: It is helpful to sketch a figure.)

The base of a railing for a staircase makes an angle of \(x\) degrees with the horizontal. Let \(d(x)\) be the horizontal distance between the two ends of the base of the railing. If point \(L\) on the railing is 5 feet higher than point \(M,\) find the positive number \(A\) such that \(d(x)=A\) cot \(x .\) Then use your function to find the length of the base of the railing if \(x=35^{\circ}\).

Find the radian measure of an angle in standard position that is generated by the specified rotation. Three full revolutions counterclockwise

Find the exact value of each expression without using a calculator. $$\tan \frac{\pi}{4} \sec \frac{\pi}{4}$$
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