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Chapter 6: Problem 8

Sketch each angle in standard position. (a) \(450^{\circ}\) (b) \(-180^{\circ}\) (c) \(270^{\circ}\)

Short Answer

Expert verified
Sketch 90掳, 180掳, and 270掳 angles starting from the positive x-axis.

Step by step solution

01

Determine the Reference Angle

For an angle, the reference angle can be found by finding the equivalent angle between 0掳 and 360掳. If the given angle is greater than 360掳, subtract 360掳 until it falls within the 0掳 to 360掳 range. If it is negative, keep adding 360掳 until it becomes positive.
02

Sketch the Angle 450掳

For 450掳, subtract 360掳 to find the equivalent angle: \[450掳 - 360掳 = 90掳\] Sketch a 90掳 angle in standard position on the coordinate plane, starting from the positive x-axis to the positive y-axis.
03

Sketch the Angle -180掳

For -180掳, add 360掳 to find the equivalent angle: \[-180掳 + 360掳 = 180掳\] Sketch a 180掳 angle in standard position, which is a straight line extending from the positive x-axis to the negative x-axis along the diameter of a circle.
04

Sketch the Angle 270掳

For 270掳, the angle directly falls between 0掳 and 360掳, hence no need for conversion. Sketch a 270掳 angle in standard position on the coordinate plane, starting from the positive x-axis to the negative y-axis.

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Key Concepts

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

Reference Angle
A reference angle is a crucial part of understanding angles on a coordinate plane. It helps simplify the process of sketching and working with angles. The reference angle is always found between the original angle and the nearest x-axis.
  • To determine the reference angle, you must bring the given angle into the standard interval of 0掳 to 360掳.
  • For angles greater than 360掳, subtract 360掳 repeatedly until it falls within 0掳 and 360掳.
  • If the angle is negative, add 360掳 repeatedly until the angle is positive.
Reference angles are useful because they give you a smaller, more manageable angle without changing the trigonometric properties relative to the initial angle. This simplifies calculations and aids in visualizing the angle on a graph.
Standard Position
In trigonometry, sketching angles is often done in what is called the 'standard position'. This method provides a consistent reference point for measuring angles. An angle is in standard position when:
  • The vertex of the angle is at the origin of the coordinate plane.
  • The initial side lies along the positive x-axis.
  • The angle is measured from the initial side, moving counterclockwise for positive angles and clockwise for negative angles.
This uniform way to draw angles allows for easy comparison and computation involving angles and shapes. It is foundational for understanding how angles interact in different mathematical contexts.
Coordinate Plane
The coordinate plane is a two-dimensional plane formed by the intersection of a vertical line (y-axis) and a horizontal line (x-axis). It is crucial when working with angles in trigonometry.
  • It supports visualizing angles by providing a framework where angles start from the origin and open outwards.
  • Aids in understanding how angles relate to each of the four quadrants, with each quadrant representing a specific range of degree measures.
  • Helps in determining the signs of trigonometric functions based on the position of an angle's terminal side.
Using the coordinate plane allows students to see the physical representation of angles, making abstract math concepts more concrete and understandable.
Positive and Negative Angles
Angles can be measured in either a positive or negative direction. This is a fundamental concept in trigonometry that affects how angles are sketched and understood.
  • Positive angles are measured counterclockwise from the initial side on the positive x-axis.
  • Negative angles are measured clockwise from the same initial side.
  • Recognizing whether an angle is positive or negative is important when determining its direction and position on the coordinate plane.
Understanding the difference between positive and negative angles helps in accurately representing angles and solving problems where angle orientations matter.

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Most popular questions from this chapter

Prove that the equations are identities. $$\cos ^{2} \theta-\sin ^{2} \theta=1-2 \sin ^{2} \theta$$

If \(a \sin ^{2} \theta+b \cos ^{2} \theta=1,\) show that $$\sin ^{2} \theta=\frac{1-b}{a-b} \quad \text { and } \quad \tan ^{2} \theta=\frac{b-1}{1-a}$$

Four functions \(S, C, T,\) and \(D\) are defined as follows: \(\left.\begin{array}{l}S(\theta)=\sin \theta \\ C(\theta)=\cos \theta \\\ T(\theta)=\tan \theta \\ D(\theta)=2 \theta\end{array}\right\\} \quad 0^{\circ}<\theta<90^{\circ}\) In each case, use the values to decide if the statement is true or false. A calculator is not required. $$S\left(45^{\circ}\right)-C\left(45^{\circ}\right)=0$$

Let \(P(x, y)\) denote the point where the terminal side of angle \(\boldsymbol{\theta}\) (in standard position) meets the unit circle (as in Figure 4). Use the given information to evaluate the six trigonometric functions of \(\theta\). \(P\) is in Quadrant I and \(x=1 / 3\)

Prove that the equations are identities. $$(1-\cos C)(1+\sec C)=\tan C \sin C$$
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