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Chapter 4: Problem 7

Draw the given angles. $$50^{\circ},-120^{\circ},-30^{\circ}$$

Short Answer

Expert verified
50掳 is counterclockwise; -120掳 and -30掳 are clockwise.

Step by step solution

01

Understanding the Positive Angle

To draw a positive angle like \(50^{\circ}\), start from the positive x-axis (the right side of the horizontal line) and rotate counterclockwise. The angle \(50^{\circ}\) means you rotate 50 degrees in the counterclockwise direction. Draw this angle in standard position starting from the positive x-axis.
02

Understanding a Negative Angle

For negative angles, such as \(-120^{\circ}\), rotation is done clockwise. Begin at the positive x-axis and rotate clockwise to reach the desired angle. For \(-120^{\circ}\), imagine starting from zero and moving clockwise 120 degrees to reach the final position of the angle.
03

Drawing the -30掳 Angle

To draw \(-30^{\circ}\), start again from the positive x-axis, and then rotate clockwise by 30 degrees. This slightly less than quarter circle brings the terminal side below the x-axis in the fourth quadrant.

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Key Concepts

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

Positive Angles
A positive angle is formed by a counterclockwise rotation starting from the positive x-axis. These angles indicate a turn in the direction opposite to that of a clock's hands. For example, a positive angle of \(50^{\circ}\) means that you begin from the right side of the horizontal axis and rotate upwards along a circular arc. Positive angles range from \(0^{\circ}\) to \(360^{\circ}\) in a single full rotation.

To draw a \(50^{\circ}\) angle:
  • Start on the positive x-axis.
  • Measure 50 degrees counterclockwise from this starting point.
  • The terminal side will be slightly above the 0-degree line in the first quadrant.
Visualizing positive angles as counterclockwise arcs makes drawing them easier and helps understand their position in a coordinate system.
Negative Angles
Negative angles involve rotation in a clockwise direction, starting from the positive x-axis. They typically indicate a rotation that goes in the same direction as a clock's hands. For example, when dealing with a negative angle like \(-120^{\circ}\), the movement is downward from the starting line.

Here's how to interpret a \(-120^{\circ}\) angle:
  • Begin at the positive x-axis direction (the familiar starting point for angle measurement).
  • Rotate clockwise for 120 degrees.
  • The terminal side of this angle will rest within the third quadrant.
Negative angles are a bit counterintuitive initially, but once you practice, you'll find them easier to visualize.
Clockwise Rotation
Clockwise rotation implies turning a figure in the direction that the hands on a clock move. For drawing angles, it's particularly relevant when dealing with negative angles. When you have an angle labeled with a negative sign, like \(-30^{\circ}\), you'll make a clockwise movement from the positive x-axis.

To create a \(-30^{\circ}\) angle:
  • Start on the positive x-axis.
  • Rotate 30 degrees in a clockwise direction.
  • This positions the terminal side in the fourth quadrant, just below the horizontal axis.
Remember that any negative angle is always coupled with a clockwise turn, which simplifies drawing and understanding.
Counterclockwise Rotation
Counterclockwise rotation points in a direction opposite to that of a clock's hands. It's the default orientation for positive angles. This means if an angle has no negative sign, it rotates counterclockwise from the positive x-axis.

Understanding a counterclockwise rotation:
  • Begins at the positive x-axis.
  • Moves upward through the quadrants based on the degree of the angle.
  • For example, a \(50^{\circ}\) angle moves slightly into the first quadrant.
This kind of rotation helps in mapping out angles accurately and is at the heart of angle measurement in trigonometry and geometry.

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Most popular questions from this chapter

Solve the given problems. Sketch an appropriate figure, unless the figure is given. A robot is on the surface of Mars. The angle of depression from a camera in the robot to a rock on the surface of Mars is \(13.33^{\circ} .\) The camera is \(196.0 \mathrm{cm}\) above the surface. How far from the camera is the rock?

Solve the given problems. Sketch an appropriate figure, unless the figure is given. When the distance from Earth to the sun is 94,500,000 mi, the angle that the sun subtends on Earth is \(0.522^{\circ} .\) Find the diameter of the sun.

Draw appropriate figures and verify through observation that only one triangle may contain the given parts (that is, any others which may be drawn will be congruent). A right triangle with a hypotenuse of \(5 \mathrm{cm}\) and a leg of \(3 \mathrm{cm}\)

Answer the given questions. If \(\tan \theta=3 / 4,\) what is the value of \(\sin ^{2} \theta+\cos ^{2} \theta ?\) \(\left[\sin ^{2} \theta=(\sin \theta)^{2}\right]\)

Solve the given problems. Sketch an appropriate figure, unless the figure is given. The headlights of an automobile are set such that the beam drops 2.00 in. for each \(25.0 \mathrm{ft}\) in front of the car. What is the angle between the beam and the road?
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