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Chapter 4: Problem 77

Given \(\sin 212^{\circ} \approx-0.53\), find another angle \(\theta\) in \(\left[0^{\circ}, 360^{\circ}\right)\) that satisfies \(\sin \theta \approx-0.53\) without using a calculator.

Short Answer

Expert verified
Another angle is \( 148^{\circ} \).

Step by step solution

01

Understand the Problem

We need to find an angle \( \theta \) such that \( \theta \in [0^{\circ}, 360^{\circ}) \) and \( \sin \theta \approx -0.53 \). We know that \( \sin 212^{\circ} = \sin \theta \), so we should look for another angle that has the same sine value within the given interval.
02

Use Symmetry of Sine Function

The sine function is periodic with a period of \(360^{\circ}\). Additionally, the sine function is negative in the third and fourth quadrants, corresponding to angles \( 180^{\circ} < \theta < 360^{\circ} \). Since \( 212^{\circ} \) is in the third quadrant, there must be an equivalent angle in the fourth quadrant.
03

Apply the Sine Symmetry Property

For any angle \( \alpha \), if \( \sin \alpha = -0.53 \), then \( \sin(360^{\circ} - \alpha) = -0.53 \) as well. Here, \( \alpha = 212^{\circ} \). Therefore, the other angle is \( 360^{\circ} - 212^{\circ} = 148^{\circ} \).
04

Verify the Angle

Let's verify the calculation: \( 360^{\circ} - 212^{\circ} = 148^{\circ} \). Indeed, \( \sin 148^{\circ} = \sin 212^{\circ} \approx -0.53 \). This confirms that our value of \( \theta \) is correct.

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Key Concepts

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

Sine Function
The sine function is a fundamental element of trigonometry. It relates the angle of a right triangle to the ratio of the opposite side over the hypotenuse. The sine function is periodic, meaning it repeats its values in regular intervals. For a complete cycle, this period is
  • 360掳 in degrees
  • or
  • \( 2\pi \) radians
The function produces values between -1 and 1. These values are derived from circles on a coordinate plane.
In the case of the angle 212掳, the sine value is approximately -0.53, indicating a location in the third quadrant of a circle where sine values are negative. Understanding these basic properties helps solve problems related to angles and their sine values.
Angle Measurement
Measuring angles can be done in degrees or radians, but degrees are more commonly used in daily contexts. An angle in this system ranges from 0掳 to 360掳 as one completes a circle.
For angle problems, understanding which quadrant an angle lies in is crucial. Quadrants are:
  • First Quadrant: 0掳 to 90掳
  • Second Quadrant: 90掳 to 180掳
  • Third Quadrant: 180掳 to 270掳
  • Fourth Quadrant: 270掳 to 360掳
The angle of 212掳 falls in the third quadrant. Here, sine values are negative, explaining why \( \sin 212^{\circ} \approx -0.53 \). Knowing this helps find other angles sharing the same sine characteristic within the 0掳 to 360掳 range.
Symmetry in Trigonometry
Symmetry plays an important role in understanding trigonometric functions. For the sine function, this symmetry is linked to the circle's properties.
Key properties include:
  • Even/Odd Nature: Sine is an odd function, meaning \( \sin(-x) = -\sin(x) \).
  • Reflections: An angle in one quadrant reflects to another quadrant while maintaining its sine value. This is due to the periodic symmetry of the circle.
In the problem, given that \( \sin 212^{\circ} \approx -0.53 \), we can use symmetry. The equivalent angle in another quadrant that reflects this property is calculated as \( 360^{\circ} - 212^{\circ} = 148^{\circ} \). Thus, \( \sin 148^{\circ} \approx -0.53 \) due to this symmetrical reflection. Understanding this concept allows us to find angles with the same trigonometric values without a calculator.

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Most popular questions from this chapter

$$ \text { Use a reference arc to determine the value of } \cos \left(\frac{29 \pi}{6}\right) \text {. } $$

For the sine and cosine functions, the domain is _________ and the range is_________.

The relationship between the coefficient \(B\), the frequency \(f\), and the period \(P\) In many applications of trigonometric functions, the equation \(y=A \sin (B t)\) is written as \(y=A \sin [(2 \pi f) t]\), where \(B=2 \pi f\). Justify the new equation using \(f=\frac{1}{P}\) and \(P=\frac{2 \pi}{B}\). In other words, explain how \(A \sin (B t)\) becomes \(A \sin [(2 \pi f) t]\), as though you were trying to help another student with the ideas involved.

In Oslo, Norway, the number of hours of daylight reaches a low of \(6 \mathrm{hr}\) in January, and a high of nearly \(18.8 \mathrm{hr}\) in July. (a) Find a sinusoidal equation model for the number of daylight hours each month; (b) sketch the graph; and (c) approximate the number of days each year there are more than \(15 \mathrm{hr}\) of daylight. Use 1 month \(\approx 30.5\) days. Assume \(t=0\) corresponds to January 1 .

In what quadrant does the angle \(t=3.7\) terminate? What is the reference arc?
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