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Chapter 4: Problem 25

In Exercises 23-26, sketch each angle in standard position. (a) \(\frac{11\pi}{6}\) (b) \(-3\)

Short Answer

Expert verified
The angle \(\frac{11\pi}{6}\) is slightly less than a full rotation anticlockwise from the positive x-axis; while the angle -3 is half a rotation clockwise from the positive x-axis.

Step by step solution

01

Sketching angle in Radians

In the first case, the angle is given in radians as \(\frac{11\pi}{6}\). The unit circle is defined such that a full rotation around the circle is \(2\pi\) radians. If you divide \(2\pi\) into 6 segments, each segment corresponds to \(\frac{\pi}{3}\) radians. So, simply sketch an angle that represents the rotation around the circle in anticlockwise direction by \(11*\frac{\pi}{3} = \frac{11\pi}{6}\) radians. This is an angle slightly less than a full rotation (which would be \(2\pi\) or approximately 6.28)
02

Sketching Negative Angle

The second angle is -3. This angle is given without a radian symbol. In this case, consider each unit as the proportion of the full rotation around the circle. As the angle is negative, the rotation is clockwise around the circle as opposed to anticlockwise for positive angles. Simply sketch an angle which represents rotation around the circle in clockwise direction by a proportion of 3/6 or half a rotation.

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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 circle with a radius of 1, centered at the origin of a coordinate plane. It is an essential tool in trigonometry and helps in understanding angles and their measures. On the unit circle:
  • The circumference is equal to the total angle measure of a circle, which is \(2\pi\) radians.
  • Angles are measured from the positive x-axis.
  • This serves as a guide for converting between radians and degrees.
The unit circle makes it easier to visualize angles and how they sweep around. Understanding the positions of angles on the unit circle is fundamental for solving trigonometric problems and sketching angles.
Radians
Radians are a way to measure angles, based on the radius of the circle. Unlike degrees, which divide a circle into 360 parts, radians use the circle's radius as the measuring stick. Here's some key information:
  • One full circle is \(2\pi\) radians.
  • Half a circle is \(\pi\) radians, which is equivalent to 180 degrees.
  • This unit connects the angle directly to the arc length on the unit circle.
Understanding radians is crucial, especially when dealing with trigonometric functions, calculus, and geometry. They provide a natural measure of angles aligning with the unit circle's geometry.
Rotation Direction
The direction in which an angle is measured determines its sign. In trigonometry:
  • Positive angles are measured anticlockwise (counterclockwise).
  • Negative angles are measured clockwise.
For example, an angle of \(\frac{11\pi}{6}\) radians is slightly less than a full rotation and is measured anticlockwise on the unit circle. Conversely, the angle \(-3\) indicates a clockwise motion, measuring effectively half a circle because a full rotation in radians is \(2\pi\), which is approximately \(6.28\). Understanding rotation directions is necessary for accurately sketching angles and solving problems.
Full Rotation
A full rotation on the unit circle corresponds to a complete sweep around the circle, returning to the starting point. This is equal to:
  • \(2\pi\) radians, or about 6.28 in decimal form.
  • 360 degrees in the degree system.
When sketching angles, knowing this helps determine whether an angle completes just a partial rotation or goes beyond a complete one. For example, an angle of \(\frac{11\pi}{6}\) radians is slightly less than \(2\pi\) and thus just under one complete rotation. This conceptual understanding simplifies the visualization and sketching of angles in trigonometric exercises.

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Most popular questions from this chapter

HARMONIC MOTION In Exercises 57-60, for the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c ) the value of \(d\) when \(t=5\), and (d) the least positive value of \(d\) for which \(t=5\). Use a graphing utility to verify your results. \(d\ =\ \dfrac{1}{2} cos\ 20 \pi t\)

In Exercises 67-76, write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example 7.) cot(arctan \(x\))

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In Exercises 99-104, fill in the blank. If not possible, state the reason. (Note: The notation \(x\rightarrow c^{+}\) indicates that \(x\) approaches \(c\) from the right and \(x\rightarrow c^{-}\) indicates that \(x\) approaches \(c\) from the left.) As \(x\rightarrow -1^{+}\), the value of arccos \(x\rightarrow\) _________.

AIRPLANE ASCENT During takeoff, an airplane's angle of ascent is \(18^\circ\) and its speed is 275 feet per second. (a) Find the plane's altitude after 1 minute. (b) How long will it take the plane to climb to an altitude of 10,000 feet?
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