Problem 37 - Algebra 2 and Trigonometry

Question

In \(28-43,\) for each function value, if \(0^{\circ} \leq \theta <3 60^{\circ},\) find, to the nearest degree, two values of \(\theta\) \(\sin \theta=0.2419\)

Step-by-Step Solution

Answer

The two values of \(\theta\) are \(14^{\circ}\) and \(166^{\circ}\).

Step 1: Understanding the Problem

We are asked to find angles \(\theta\) within the range \(0^{\circ} \leq \theta < 360^{\circ}\) for which the sine of the angle equals 0.2419. We need two such values, rounded to the nearest degree.

Step 2: Finding the First Solution

First, we take the inverse sine of 0.2419 to calculate one of the values of \(\theta\). Using a calculator, \(\theta = \sin^{-1}(0.2419) \approx 14^{\circ}\).

Step 3: Determining the Second Solution

Since the sine function is positive in the first and second quadrants, and has a period of \(360^{\circ}\), we find the second angle by subtracting the first value from \(180^{\circ}\): \(\theta_{2} = 180^{\circ} - 14^{\circ} = 166^{\circ}\).

Step 4: Checking the Range

Both calculated angles, \(14^{\circ}\) and \(166^{\circ}\), lie within the specified range of \(0^{\circ} \leq \theta < 360^{\circ}\). Thus, both are valid solutions.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions. It relates the angle of a right triangle to the ratio of the length of the side opposite the angle to the hypotenuse. Simply put, it helps us understand the relationship between angles and side lengths in a triangle.
This is represented formulaically as \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\).

The sine function is periodic, with a wave-like pattern that repeats every \(360^{\circ}\) or \(2\pi\) radians.
This periodic nature is why sine values repeat for different angles as you move through the cycle.

• The sine of \(0^{\circ}\) is \(0\).
• The sine of \(90^{\circ}\) is \(1\).
• The sine of \(180^{\circ}\) is back to \(0\).
• The sine of \(270^{\circ}\) is \(-1\).

These periodic values help us find equivalent sine values for different angles in different cycles.

Inverse Sine

The inverse sine function, often written as \(\sin^{-1}\) or arcsin, is used to find the angle that corresponds to a given sine value.
When you know the sine of an angle, you can use the inverse sine to find out what that angle is.

For example, given that \(\sin \theta = 0.2419\), we use the inverse sine to calculate the angle \(\theta\).
Using a calculator, you would input \(\sin^{-1}(0.2419)\), which gives \(\theta \approx 14^{\circ}\).

This function is crucial in trigonometry for solving problems where the angle needs to be determined from a sine value.

• Inverse sine is limited to angles between \(-90^{\circ}\) and \(90^{\circ}\), which helps in determining angles in the first and fourth quadrants.
• This constraint sometimes necessitates additional steps to find all possible angle solutions when dealing with analytical problems like the original exercise.

Angle Measurement

Angles in mathematics are usually measured in degrees or radians. Here, we focus on degree measurement as it is common in introductory trigonometry.
Angles tell us how much one direction simulates another.

There are 360 degrees in a full circle.
Each degree is further divided into 60 minutes, and each minute into 60 seconds.

This division helps in precise measurements, as angles are fundamental in various applications.

• One degree is \(1/360\) of the full circle.
• Angles go counterclockwise from the positive x-axis, which is the standard practice in trigonometry.

Understanding angle measurement is key to solving geometry and trigonometry problems, like determining equivalent angle positions.

Quadrants in Trigonometry

The Cartesian plane is divided into four quadrants, each helping in interpreting angle locations and sign values of trigonometric functions.
These quadrants illustrate where angles can ``live'' based on their sine or cosine values.

• First Quadrant: Angles from \(0^{\circ}\) to \(90^{\circ}\). All trigonometric functions are positive.
• Second Quadrant: Angles from \(90^{\circ}\) to \(180^{\circ}\). The sine function remains positive, while cosine and tangent are negative.
• Third Quadrant: Angles between \(180^{\circ}\) and \(270^{\circ}\). Both sine and cosine are negative, while tangent is positive.
• Fourth Quadrant: Angles from \(270^{\circ}\) to \(360^{\circ}\). Here, sine is negative, cosine is positive, and tangent is negative.

By understanding these quadrants, one can deduce the signs of sine, cosine, and tangent functions, which simplifies the process of identifying possible angle solutions.