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Chapter 7: Problem 43

Find the reference angle of each angle. $$ 120^{\circ} $$

Short Answer

Expert verified
The reference angle is \(60^{\text{鈭榼}\).

Step by step solution

01

Identify the Quadrant

The first step is to determine in which quadrant the given angle lies. The angle given is \(120^{\text{鈭榼}\), which lies in the second quadrant (since \(90^{\text{鈭榼} < 120^{\text{鈭榼} < 180^{\text{鈭榼}\)).
02

Calculate the Reference Angle

To find the reference angle in the second quadrant, use the formula: \[ \theta_{\text{ref}} = 180^{\text{鈭榼} - \theta \ \text{where } \theta = 120^{\text{鈭榼} \]Thus: \[ \theta_{\text{ref}} = 180^{\text{鈭榼} - 120^{\text{鈭榼} = 60^{\text{鈭榼} \]

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Key Concepts

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

Quadrant Identification
Understanding in which quadrant an angle lies is essential in trigonometry. Each quadrant in the coordinate system helps us know the nature of the trigonometric functions of angles.

The coordinate system has four quadrants:
  • First Quadrant: angles from 0掳 to 90掳
  • Second Quadrant: angles from 90掳 to 180掳
  • Third Quadrant: angles from 180掳 to 270掳
  • Fourth Quadrant: angles from 270掳 to 360掳

    • When given an angle like 120掳, you should first figure out its position in the coordinate system. Since 120掳 lies between 90掳 and 180掳, it is in the Second Quadrant.

    In the Second Quadrant, the reference angle is always subtracted from 180掳. This step is crucial as different quadrants have different rules for calculating reference angles.
Trigonometry
Trigonometry deals with the relationships between the angles and the sides of triangles. In this context, understanding the reference angle helps simplify problem-solving tasks involving trigonometric functions.

Key terms include:
  • Sine (sin)
  • Cosine (cos)
  • Tangent (tan)
  • Reference Angle

    • Using the reference angle can help you easily find trigonometric values. It serves as a simplified measure since it is the acute angle (less than 90掳) formed by the terminal side of the angle and the x-axis. This approach is beneficial as it maintains the values within the more easily manageable range of 0掳 to 90掳.
Angle Measurement
Angles can be measured in degrees (掳) or radians. For most basic trigonometry problems, degrees are used. Understanding how to calculate reference angles involves straightforward subtraction.

With an angle of 120掳, we already determined it lies in the Second Quadrant. Here, to find the reference angle, use the subtraction formula:

\[ \theta_{\text{ref}} = 180^{\text{鈭榼} - \theta \]
\[ \theta_{\text{ref}} = 180^{\text{鈭榼} - 120^{\text{鈭榼} = 60^{\text{鈭榼} \]

Thus, the reference angle is 60掳. This value simplifies our trigonometric computations significantly. Using this method ensures consistency and accuracy in solving trigonometric problems.

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Most popular questions from this chapter

Given \(\sec \theta=3,\) use trigonometric identities to find the exact value of (a) \(\cos \theta\) (b) \(\tan ^{2} \theta\) (c) \(\csc \left(90^{\circ}-\theta\right)\) (d) \(\sin ^{2} \theta\)

Name the quadrant in which the angle \(\theta\) lies. $$ \sec \theta<0, \quad \tan \theta>0 $$

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$\cot 40^{\circ}-\frac{\sin 50^{\circ}}{\sin 40^{\circ}}$$

A point on the terminal side of an angle \(\theta\) in standard position is given. Find the exact value of each of the six trigonometric functions of \(\theta .\) $$ \left(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right) $$

Convert each angle in radians to degrees. Express your answer in decimal form, rounded to two decimal places. 3
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