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Chapter 5: Problem 9

Sketch the angles in standard position. $$135^{\circ}$$

Short Answer

Expert verified
The angle in standard position is drawn starting from the positive x-axis and rotating counterclockwise till it forms an angle of 135 degrees with the positive x-axis.

Step by step solution

01

Draw the x and y axes

Draw two perpendicular lines intersecting at their midpoints. Label the horizontal line as 'x-axis' and the vertical one as 'y-axis'.
02

Indicate the Initial Side

Indicate the initial side of the angle on the positive x-axis. This is the starting position of the angle.
03

Draw the Angle

From the initial side, rotate a line counterclockwise by 135 degrees. The rotated line represents the terminal side of the angle. This forms an angle of 135 degrees with the positive x-axis.

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Key Concepts

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

Standard Position
Whenever you sketch or describe an angle, putting it in standard position is crucial. In this setup, you start by defining a fixed point at the origin of a coordinate system 鈥� where the x-axis and y-axis intersect. This ensures that all angles begin from a common starting point, making them easy to understand and compare.

The positive x-axis is set as the reference line, from where the angle's measure begins. An angle measured in standard position typically rotates counterclockwise from this reference line for positive angles. Setting angles in standard position is important because it:
  • Makes it simple to identify and measure angles
  • Allows consistent comparison between different angles
  • Creates a standard approach for interpreting geometry problems
Initial Side
The initial side of an angle is where everything begins! It's the starting point on which the angle is based. In standard position, this initial side is always along the positive x-axis.

Here's what makes the initial side significant:
  • It represents the zero degree starting point for the angle.
  • Understanding its location helps in determining the direction and the measure of the angle accurately.
By fixing the initial side on the positive x-axis, angles can easily be sketched and identified, especially when dealing with problems that require precise angle measurement like trigonometric functions.
Terminal Side
The terminal side of an angle is the side where the angle "finishes" or "ends". After rotating from the initial side, this line marks the final boundary of the angle. For the angle to truly be in standard position, the rotation occurs from the positive x-axis.

Once you establish the terminal side, you can fully represent angles such as the given 135掳, which involve rotating counterclockwise to form the desired measure. Key points to note about the terminal side include:
  • It determines the angle's magnitude by showing how far the rotation extends from the initial side.
  • Understanding it allows one to place the angle correctly in space, which is vital when sketching angles.
By identifying both its position and orientation, the terminal side completes the angle, helping you visualize and solve geometry and trigonometry problems effectively.

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Most popular questions from this chapter

Graph at least two cycles of the given functions. $$g(x)=\frac{1}{2} \sin (2 x-\pi)$$

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

Consider an angle \(\theta\) in standard position whose vertex coincides with the center of a circle of radius \(r .\) The portion of the circle bounded by the initial side and the terminal side of the angle \(\theta\) is called a sector of the circle. (a) If \(A\) is the area of the circle, then \(A_{s}=A \frac{\theta}{2 \pi}\) represents the area of the sector because \(\frac{\theta}{2 \pi}\) gives the fraction of the area covered by the sector. Show that the area of a sector, \(A_{s},\) is \(A_{s}=\frac{r^{2} \theta}{2} .\) Here theta is in radians. (b) Find \(A_{s}\) if \(\theta=\frac{\pi}{3}\) and \(r=12\) inches.

Graph at least two cycles of the given functions. $$h(x)=2 \cos \left(2 x+\frac{\pi}{2}\right)-1$$

Use the negative-angle identities to compute the exact value of each of the given trigonometric functions. $$\tan \left(-\frac{7 \pi}{3}\right)$$
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