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Chapter 4: Problem 26

Sketch each angle in standard position. (a) 4 (b) \(7 \pi\)

Short Answer

Expert verified
The angle 4 in standard position originates at the positive x-axis and rotates 4 degrees counter-clockwise. The angle of \( 7 \pi \) radians in standard position goes 3.5 times around the circle and stops at the negative x-axis.

Step by step solution

01

Identify Angle Type

First, identify the type of angle that is given. If it is a multiple of \( \pi \), it means the angle is presented in radians. If there is no \( \pi \) involved, assume it's given in degrees.
02

Convert To Suitable Units If Needed

Convert the angle to the unit you're comfortable with if necessary. In this case, both angles are already expressed in their simplest forms and can therefore be straightaway sketched on a graph. For instance, if an angle given in radians was to be converted to degrees, the conversion factor of \( \frac{180^{\circ}}{\pi radians} \) would be used and vice versa.
03

Sketch Angle 4 in Standard Position

Standard position for an angle of 4 degrees is a slight rotation counter-clockwise from the positive x-axis. Remember the rotation is counter-clockwise if the angle is positive and clockwise if the angle is negative.
04

Sketch Angle \( 7 \pi \) in Standard Position

An angle of \( 7 \pi \) radians means the angle has revolved 7 half turns about the origin. Because each half turn is \( \pi \) radians or 180 degrees, 7 half turns gives us 3 full turns and a half, or 1,260 degrees. This will have us end up on the negative x-axis, since a half turn from the positive x-axis (where we start) lands us on the negative x-axis.
05

Review Your Sketches

Take a moment to review your sketches and ensure they truly represent the given angles in standard positions. From your sketch, angle 4 should be a slight tilt from the positive x-axis counter-clockwise and angle \( 7 \pi \) should be landing on the negative x-axis.

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