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Chapter 2: Problem 28

Sketch the angles with given measure in standard position. $$ -150^{\circ} $$

Short Answer

Expert verified
The angle \(-150^{\circ}\) is positioned to be halfway between the negative x-axis and the negative y-axis in quadrant III.

Step by step solution

01

Determine Angle Direction

The given angle is \(-150^{\circ}\). Since the degree is negative, this means the angle is measured clockwise from the positive x-axis.
02

Locate Initial Side

In standard position, the initial side of an angle is always the positive x-axis. This is where the measurement begins.
03

Measure the Angle Clockwise

To sketch the angle \(-150^{\circ}\), move clockwise from the positive x-axis. Remember that a full rotation clockwise from the positive x-axis is \(-360^{\circ}\). Since \(-150^{\circ}\) is less than this, it will be positioned before completing the half rotation \(-180^{\circ}\).
04

Sketch the Final Position

Starting at the positive x-axis (the initial side), rotate clockwise to approximately halfway between \(-90^{\circ}\) and \(-180^{\circ}\). This is where \(-150^{\circ}\) lies.

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Key Concepts

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

Standard Position
In angle sketching, the concept of 'standard position' is fundamental to understanding how angles are represented on a coordinate plane. When an angle is in standard position, its initial side is always placed along the positive x-axis. This means that every angle begins its measurement from this horizontal line lying to the right. The terminal side, or where the angle points, can rotate either clockwise or counterclockwise from the initial side, depending on whether the angle is negative or positive.
  • This setup allows for simplicity and consistency, playing a key role in correctly interpreting and sketching angles.
  • Standard position also helps in visualizing and calculating trigonometric functions as they directly relate to the position of points on the circle derived from rotating the terminal side.

Understanding standard position gives you a solid starting point for tackling any angle-related problem, ensuring accurate sketches and calculations every time.
Clockwise Measurement
Measuring angles in a clockwise direction often indicates the angle has a negative measure. This can initially be tricky for students to grasp, because we usually think of rotation as positive when moving counterclockwise. However, in mathematics, traveling clockwise from the initial side on the positive x-axis results in a negative angle measure.
  • A clockwise rotation means you are moving in the same direction as the hands of a clock.
  • In the case of our example, an angle of \( -150^{\circ} \) illustrates this "clockwise" movement very clearly.

So how do you apply this? The key is to imagine turning your arm around the central point (the origin), starting from the positive x-axis and swinging downwards to your desired degree. This is essential for accurately sketching the angle and understanding the spatial orientation.
Initial Side
The 'Initial Side' of an angle in standard position is the baseline from which we start measuring. It's consistently the positive x-axis, making it a stable and predictable starting line. Knowing the initial side is crucial as it provides a fixed reference point.
  • This fixed line helps ensure that no matter what angle you're working with, the start point is the same.
  • This is particularly significant when dealing with rotated angles or calculating precise positions.

So, when sketching or calculating angles, always remember that the initial side is your ruler edge鈥攕teady and unchanging. Every angle's journey starts here, setting the stage for accurate rotations and measurements.

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Most popular questions from this chapter

Use a reciprocal identity to find the function value indicated. Rationalize denominators if necessary. If \(\cot \theta=-\frac{\sqrt{7}}{5}\), find \(\tan \theta\)

Find the smallest possible positive measure of \(\theta\) (rounded to the nearest degree) if \(\cos \theta=-0.2388\) and the terminal side of \(\theta\) (in standard position) lies in quadrant III. Solution: Evaluate with a calculator. $$ \theta=\cos ^{-1}(-0.2388)=103.8157^{\circ} $$ Approximate to the nearest degree. \(\quad \theta \approx 104^{\circ}\) This is incorrect. What mistake was made?

Evaluate the following expressions exactly by using a reference angle. $$ \sin 315^{\circ} $$

Find the smallest possible positive measure of \(\theta\) (rounded to the nearest degree) if the indicated information is true. \(\tan \theta=-0.8391\) and the terminal side of \(\theta\) lies in quadrant II.

Find the smallest possible positive measure of \(\theta\) (rounded to the nearest degree) if the indicated information is true. \(\sin \theta=-0.4226\) and the terminal side of \(\theta\) lies in quadrant III.
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