Problem 41 Find all angles in the interval ... [FREE SOLUTION]

91影视

Trigonometry

Mark Dugopolski

$Math Studyset 91影视 Explanations$ Math

5 Edition

Chapter 4: Problem 41

Find all angles in the interval $\left[0^{\circ}, 360^{\circ}\right]$ that satisfy each equation. Round approximations to the nearest tenth of a degree. $$ \sin \alpha=-0.244 $$

Short Answer

Expert verified

The angles are 194.1掳 and 345.9掳.

Step by step solution

Understand the sine function properties

The sine function has a periodic nature with a period of 360掳. It also has symmetry properties: \ - \ For $ \theta $ in the first and fourth quadrants: $ \text{sin}(\theta) = -0.244 $

Find the reference angle

Use the inverse sine function to find the reference angle. \ Calculate: \ $ \theta = \text{sin}^{-1}(0.244) $. \ Using a calculator \ $ \theta \approx \ 14.1掳 $

Determine possible angles

Since sine is negative in the third and fourth quadrants, the possible angles are: \ $180掳 + \theta $ and \ $360掳 - \theta $ for \ \theta = 14.1掳 \ \ These angles are: \ \ $ 180掳 + 14.1掳 = 194.1掳 $ $ 360掳 - 14.1掳 = 345.9掳 $

Conclusion of angles within interval

Therefore, the angles satisfying $\sin \alpha = -0.244 $ in the given interval are \ 194.1掳 \ and \ 345.9掳 \

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Key Concepts

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

Sine Function Properties

The sine function, noted as $\text{sin} $, is a critical concept in trigonometry. It relates an angle of a triangle to the ratio of the length of the opposite side over the hypotenuse. This function is periodic with a period of \[360^{\text{掳}}\], which means its values repeat every 360掳. Additionally, sine has symmetry properties: in the first quadrant, sine values are positive; in the second quadrant, sine values are also positive but decrease; in the third and fourth quadrants, sine values are negative. These quadrants help determine the sign of the sine value when solving trigonometric equations.

Inverse Sine Calculation

The inverse sine function, written as \[ \text{sin}^{-1} \], or \[ \text{arcsin} \], is used to find an angle that yields a particular sine value. Suppose we need to find an angle \[ \theta \] such that \[ \text{sin}(\theta) = -0.244 \]. First, calculate the inverse sine of the absolute value: \[ \theta = \text{sin}^{-1}(0.244) \]. Using a calculator, we find \[ \theta \] to be approximately \[ 14.1^{\text{掳}} \]. It's important to remember that the inverse sine function typically outputs values in the range \[ -90^{\text{掳}} \] to \[ 90^{\text{掳}} \].

Angle Determination in Quadrants

Since \[ \text{sin}\theta = -0.244 \], we need to locate \[ \theta \] angles where sine is negative. Sine is negative in the third and fourth quadrants:

In the third quadrant, the angle is \[ 180^{\text{掳}} + \theta \]
In the fourth quadrant, the angle is \[ 360^{\text{掳}} - \theta \]

For our case, we use \[ \theta = 14.1^{\text{掳}} \]:

Third quadrant angle: \[ 180^{\text{掳}} + 14.1^{\text{掳}} = 194.1^{\text{掳}} \]
Fourth quadrant angle: \[ 360^{\text{掳}} - 14.1^{\text{掳}} = 345.9^{\text{掳}} \]

So, the angles within the interval \[ \text{0掳 to 360掳} \] that satisfy \[ \text{sin}\theta = -0.244 \] are \[ 194.1^{\text{掳}} \] and \[ 345.9^{\text{掳}} \].

Reference Angle

A reference angle is the acute angle formed by the terminal side of a given angle with the horizontal axis. When solving trig equations, use the absolute value of the inverse sine to find this acute angle. In our example, the reference angle for \[ \text{sin}\theta \] of \[ -0.244 \] is \[ 14.1^{\text{掳}} \]. This reference angle is then used to determine the actual angles in their respective quadrants. Understanding reference angles helps in tackling more complex trigonometric problems effectively.

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91影视

Find all angles in the interval \(\left[0^{\circ}, 360^{\circ}\right]\) that satisfy each equation. Round approximations to the nearest tenth of a degree. $$ \sin \alpha=-0.244 $$

Short Answer

Step by step solution

Understand the sine function properties

Find the reference angle

Determine possible angles

Conclusion of angles within interval

Key Concepts

Sine Function Properties

Inverse Sine Calculation

Angle Determination in Quadrants

Reference Angle

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