Problem 24 - Precalculus : Building Concepts and Connections

Question

Use a calculator to evaluate each trigonometric function. Make sure that the calculator is in \(R A D I A N\) mode. $$\sin ^{-1} 0.8$$

Step-by-Step Solution

Answer

The inverse sine of 0.8, calculated in Radian mode, is approximately 0.927 radians.

Step 1: Set Calculator to Radian Mode

Before performing any calculations, ensure your calculator is set to RADIAN mode. This is necessary for accurate results when dealing with trigonometric functions.

Step 2: Input Inverse Sine Function

Now you're ready to input the problem into your calculator. Enter \(\sin ^{-1} 0.8\), which asks for the inverse sine of 0.8. This should return a numerical value.

Step 3: Interpret the Result

The resulting number is the measure, in radians, of the angle whose sine is 0.8. This is the solution to the problem.

Key Concepts

Radians

Radians are a way to measure angles, much like degrees. In fact, radians are based on the properties of circles and their radius. When you think of a circle, the radius is the distance from the center to any point on its edge. If you imagine wrapping the radius along the circumference of the circle, the angle subtended by that arc is one radian.

Think of radians as a natural fit for dealing with angles in trigonometry. This is because radians relate more closely to the mathematics of periodic functions like sine and cosine, which are fundamental in trigonometry. When your calculator is in radian mode, it uses this connection to give more precise results, especially important in scientific calculations.

• 1 radian is about 57.3 degrees, but typically, calculations are done in terms of \( \pi \), so a full circle is \( 2\pi \) radians.
• For computations involving inverse trigonometric functions, ensure your calculator is set to radians to interpret angle measures correctly.

Trigonometric Functions

Trigonometric functions are critical in understanding relationships between angles and sides of triangles, especially right-angled ones. They include sine, cosine, and tangent, each with its unique properties and uses.

These functions have an extension known as inverse trigonometric functions, like the inverse sine (sin\(^{-1}\)), inverse cosine (cos\(^{-1}\)), and inverse tangent (tan\(^{-1}\)). The inverse sine function specifically allows us to find an angle that has a specific sine value.

• The usual range for \( \sin^{-1}(x) \) is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).
• Inverse functions are used when you have the sine value and need to determine the angle itself.

Knowing how to determine or solve using these trigonometric functions is a huge advantage in understanding periodic phenomena like waves and oscillations.

Mathematical Problem Solving

Solving mathematical problems often involves using functions and understanding their properties. With trigonometric problems, like finding the angle with a specified sine, the process involves strategic use of a calculator or mathematical software.

• Begin by correctly setting up your tool, ensuring it is in the appropriate mode (radians).
• Input the known values into the function or inverse function as required.
• Interpret the output correctly, considering unit measures like radians.

When facing a problem, double-check if your calculator settings are correct to avoid erroneous results. Practical applications of these solutions span across physics, engineering, and various technological domains. The key is ensuring you understand each step and translate the mathematical results back into real-world meaning if necessary.