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Chapter 12: Problem 10

Find the measure of each angle accurate to the nearest degree. $$\sin A=0.5$$

Short Answer

Expert verified
The angles are 30掳 and 150掳.

Step by step solution

01

Identify the basic equation

The given trigonometric equation is \(\text{sin } A = 0.5\). Our goal is to find the angle \(A\) that satisfies this equation.
02

Determine the principal angle

Use the inverse sine function to determine the principal angle. This is done by calculating \(\text{A} = \text{sin}^{-1}(0.5)\).
03

Calculate the angle using a calculator

Using a scientific calculator, find \( \text{A} = \text{sin}^{-1}(0.5) \). This gives \( \text{A} = 30^\text{o} \).
04

Identify all possible angles

Since the sine function is positive in both the first and second quadrants, the possible angles are \( A = 30^\text{o} \) and \( A = 180^\text{o} - 30^\text{o} = 150^\text{o} \). Therefore, the two angles that satisfy \( \text{sin} A = 0.5 \) are \( 30^\text{o} \) and \( 150^\text{o} \).

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Key Concepts

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

Inverse Sine
Inverse sine, also known as arcsine and denoted as \(\text{sin}^{-1}\), is a function that helps us find the angle when the sine value is known. In simpler terms, if we know that \(\text{sin} A = 0.5\), we can use the inverse sine function to find the angle \(\text{A}\). This is done through the equation \(\text{A} = \text{sin}^{-1}(0.5)\).
Inverse sine values range from \(-90^\text{o}\) to \(90^\text{o}\) (or \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) in radians). This restriction is important because it ensures that each sine value corresponds to a single angle, making calculations straightforward and unambiguous.
Principal Angle
The principal angle is the smallest positive angle that satisfies the given trigonometric equation. For the sine function, it is typically in the range from \(0^\text{o}\) to \(180^\text{o} \) (or \(0\pi\) to \((\pi\)) radians).
When we use the inverse sine function to find our angle, the result is always the principal angle within this range. For instance, when we calculate \(\text{A} = \text{sin}^{-1}(0.5)\), we find that \(\text{A}\) is \(30^\text{o}\).
This principal angle is crucial because it acts as a starting point for finding all other angles that satisfy the original equation.
Angle Calculation
Calculating angles when given a sine value involves a few key steps:
    1. Use the inverse sine function to find the principal angle. For \( \text{sin} A = 0.5 \), this gives \( A = 30^\text{o} \).
    2. Determine in which quadrants the sine value is positive or negative. The sine of an angle is positive in the first and second quadrants.
    3. To find other possible angles when \(\text{sin} A = 0.5\), use the symmetry of the sine function. Besides \((30^\text{o})\), an additional angle in the second quadrant is computed as: \text{180^\text{o} - 30^\text{o} = 150^\text{o}}
    Ultimately, the two angles that meet \(\text{sin} A = 0.5\) are \(30^\text{o}\) and \(150^\text{o}\).
This process ensures all possible solutions are accounted for, giving a comprehensive understanding of how to solve trigonometric equations.

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Most popular questions from this chapter

A lighthouse 55 meters above sea level spots a distress signal from a sailboat. The angle of depression to the sailboat measures \(21^{\circ} .\) How far away is the sailboat from the base of the lighthouse?

Use a calculator to find each trigonometric ratio accurate to the nearest ten thousandth. $$\cos 29^{\circ}$$

Solve for \(x\). Express each answer accurate to the nearest hundredth of a unit. $$\cos 52^{\circ}=\frac{19}{x}$$

Application A lighthouse is east of a Coast Guard patrol boat. The Coast Guard station is \(20 \mathrm{km}\) north of the lighthouse. The radar officer aboard the boat measures the angle between the lighthouse and the station to be \(23^{\circ} .\) How far is the boat from the station?

Draw a regular decagon with side length \(6 \mathrm{cm}\) Divide the decagon into ten congruent isosceles triangles. a. Find the length of the apothem, and then use the Regular Polygon Area Conjecture to find the area of the decagon. b. Find the lengths of the legs of the isosceles triangles, and then use the SAS Triangle Area Conjecture to find the area of each isosceles triangle. Multiply to find the area of the decagon. c. Compare your answers from parts a and b.
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