Problem 17 - Algebra 2 and Trigonometry

Question

In \(13-24,\) find, to the nearest ten-thousandth, the radian measure \(\theta\) of a first-quadrant angle with the given function value. $$ \sin \theta=0.8267 $$

Step-by-Step Solution

Answer

The radian measure \(\theta\) is approximately 0.9735.

Step 1: Understanding the Function

We are given that the sine of angle \(\theta\) is 0.8267. This means \(\sin \theta = 0.8267\). Our task is to find the angle \(\theta\) in radians.

Step 2: Determine the Quadrant

The problem specifies that \(\theta\) is a first-quadrant angle. In the first quadrant, all trigonometric function values are positive, including sine.

Step 3: Use Inverse Sine

To find \(\theta\), we need to use \(\sin^{-1}\) or arcsin to determine the angle whose sine value is 0.8267. Use a calculator to find \(\theta = \sin^{-1}(0.8267)\).

Step 4: Conversion to Radians

Ensure the calculator is set to radians mode. Compute \(\sin^{-1}(0.8267)\) and ensure the result is given in radians.

Step 5: Rounding to the Nearest Ten-Thousandth

After calculating \(\theta\), round the value to four decimal places (nearest ten-thousandth). If \(\sin^{-1}(0.8267)\approx 0.9735\), \(\theta\) rounded to the nearest ten-thousandth is 0.9735.

Key Concepts

Understanding Radian Measure

Radian measure is a fundamental concept in trigonometry, used to describe angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians are based on the circle's radius. One radian is the angle created when the arc length is equal to the radius of the circle. Since the total circumference of a circle is \(2\pi\) times its radius, a full circle is \(2\pi\) radians. This makes one complete revolution equivalent to \(360^\circ\), which is \(2\pi\) radians. Consequently, \(180^\circ\) is \(\pi\) radians, and \(90^\circ\) is \(\pi/2\) radians.To convert between degrees and radians, you can use the formula:

• Degrees to Radians: \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \)
• Radians to Degrees: \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \)

Understanding radian measure is crucial for working with trigonometric functions, as many scientific calculators and mathematical software require input in radians rather than degrees.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles when given a trigonometric ratio. For the sine function, its inverse is known as arcsine, denoted by \( \sin^{-1} \) or \( \arcsin \).When we have the sine of an angle, such as \( \sin \theta = 0.8267 \), using \( \arcsin \) helps determine the angle \( \theta \) whose sine is this value. It's important to remember that inverse trigonometric functions typically return results in a specific range to keep them unique. For \( \sin^{-1} \), the output (angle \( \theta \)) typically lies between \(-\frac{\pi}{2} \) and \( \frac{\pi}{2} \), unless specified by the context, such as a first-quadrant angle.To find \( \theta \) in radians where \( \sin \theta = 0.8267 \), you would compute \( \theta = \sin^{-1}(0.8267) \) using a calculator set to radian mode. This helps in determining the exact measure of angle \( \theta \), reflecting how inverse trigonometric functions link ratios back to angles.

First Quadrant Angles

In trigonometry, the first quadrant of the coordinate plane is the region where both the x and y coordinates are positive. This quadrant is particularly significant because all basic trigonometric function values (sine, cosine, tangent, etc.) are positive here.An angle in the first quadrant is any angle between \(0\) and \(\frac{\pi}{2}\) radians (or between \(0^\circ\) and \(90^\circ\)). This context allows for straightforward computation since no negative signs or other quadrant rules affect the trigonometric values.When problems specify a first-quadrant angle, it gently reminds us that:

• All standard trigonometric functions return positive results.
• Calculators in either degrees or radians can compute the arcsine value directly, confident that these results are meaningful for this range.

Knowing that \( \theta \) is a first-quadrant angle helps narrow down the potential solutions and confirm the correctness of computed results. Understanding these characteristics ensures clarity when working with trigonometric functions.