Problem 22 - Algebra 2 and Trigonometry

Question

In \(15-23,\) use a calculator to find each value of \(\theta\) to the nearest degree. $$ \theta=\arcsin (-0.72) $$

Step-by-Step Solution

Answer

\( \theta \approx -46^\circ \).

Step 1: Understand the Problem

We need to find the angle \( \theta \) in degrees for which \( \sin(\theta) = -0.72 \). Since the function here is \( \arcsin \), we are looking for the inverse sine of \(-0.72\).

Step 2: Use a Calculator

Set the calculator to degree mode. Use the calculator to find \( \theta = \arcsin(-0.72) \) which will give you the angle \( \theta \) corresponding to the sine of \(-0.72\).

Step 3: Interpret the Calculator's Output

The calculator will provide an angle within the range of \( -90^\circ \) to \( 90^\circ \) (or \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) in radians) because the arcsine function only outputs angles in this range. Review the output to ensure it meets the context of the problem.

Step 4: Round the Result

Once the calculator gives you an answer, which is likely a decimal, round this value to the nearest degree as the problem asks for the angle to be in whole degrees.

Key Concepts

Arcsin Function

The arcsin function, also known as the inverse sine function, helps us find an angle when we know its sine value. In mathematical notation, it is represented as \( \arcsin(x) \).
This function is crucial when working with trigonometry since it allows us to reverse-engineer angles from known sine values. The sine function gives us the ratio of the opposite side to the hypotenuse in a right triangle, while the arcsin function does the opposite – it takes this ratio and provides the corresponding angle.

• Range of Arcsin: The arcsin function only outputs angles from \(-90^\circ\) to \(90^\circ\). This is because the sine of an angle in that range will cover all possible values from \(-1\) to \(1\).
• Practical Use: Use the arcsin function whenever you need to find an angle based on its sine value, particularly when solving triangles or analyzing periodic phenomena.

Degree Mode

When using a calculator to find inverse trigonometric functions like arcsin, it's essential to ensure the device is in degree mode. Calculators can operate in different modes, with degree and radian being most common. Degree mode will interpret your calculations based on a 360-degree circle, which is typical in many everyday applications like navigation and rotation.

• Checking the Mode: Before starting your calculations, make sure the calculator displays a 'DEG' or '°' symbol indicating it's set to degree mode.
• Why Degree Mode: Since the exercise specifically asks for the angle in degrees, using this mode ensures that your results align with the problem requirements.

Rounding Angles

Rounding angles is a common step in trigonometry problems to simplify results and make them more practical for interpretation. After calculating an angle, especially when using inverse trigonometric functions like arcsin, the output is usually a decimal.
Rounding this decimal to the nearest whole number makes it easier to use and understand in real-world contexts, where exact fractions of degrees are often unnecessary or impractical.

• Rounding Process: Look at the first decimal point. If it is 5 or greater, round the angle up, otherwise round down.
• Application: For the problem \(\theta = \arcsin(-0.72)\), once you have this in decimal form, rounding gives you a whole number degree angle to report as your answer.