Problem 50 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to give each value of \(\theta\) in decimal degrees. $$\theta=\arcsin 0.77900016$$

Step-by-Step Solution

Answer

\( \theta \approx 51.228 \) degrees.

Step 1: Understanding the Inverse Sine Function

The problem involves determining the angle \( \theta \) whose sine value is 0.77900016. This means we are looking for \( \theta \) such that \( \sin(\theta) = 0.77900016 \). This is done using the inverse sine function, also known as arcsine, denoted as \( \arcsin \).

Step 2: Using a Calculator to Find \( \theta \)

To find \( \theta \), use a scientific calculator with an inverse sine function. Enter the value 0.77900016 into the calculator and then apply the \( \arcsin \) function to obtain \( \theta \). Ensure the calculator is set to provide the result in degrees, not radians.

Step 3: Note the Result

The calculator should provide the result as \( \theta \approx 51.228 \) degrees. This means the angle whose sine is 0.77900016 is approximately 51.228 degrees. Ensure to check that the calculator is in degree mode to ensure accuracy.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles of a right triangle to the ratios of its sides. Among these functions are sine, cosine, and tangent, represented as \( \sin \), \( \cos \), and \( \tan \) respectively. They are primarily used in geometry, engineering, and physics. The sine function, for example, is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. Inverse trigonometric functions come into play when we need to determine an angle given its trigonometric ratio. In our exercise, we specifically use the inverse sine function, known as \( \arcsin \). This function helps us find an angle whose sine is a given number. Hence, given a sine value, the \( \arcsin \) function will return the corresponding angle (\( \theta \)) in a specified unit (degrees or radians). It's essential to grasp this foundational concept of trigonometry, as it connects numerical values to geometric angles, enabling a deeper understanding of spatial relationships.

Scientific Calculator Usage

Scientific calculators are indispensable tools for dealing with trigonometric functions and their inverses. When using a calculator to determine an angle from a sine value, you'll employ the inverse sine (\( \arcsin \)) function. Here’s a simplified guide on effectively using a scientific calculator for this purpose:

• First, ensure your calculator is turned on and ready to operate.
• Look for the "arcsin" button or function. This might require a "shift" or "2nd" function key on some calculators.
• Input the given sine value, 0.77900016, in this case.
• Press the "arcsin" button to calculate the angle \( \theta \).
• Make sure your calculator is in the proper mode (degrees or radians) that you desire.

With this step-by-step approach, you should easily find the angle related to any set sine value. It's also important to understand your calculator’s display and settings well, as incorrect mode settings will lead to wrong answers.

Decimal Degrees

In trigonometry, angles are often measured in degrees, and here, we use decimal degrees for precision. Unlike using whole numbers or fractions, decimal degrees allow for exactness, which is crucial in scientific computations. Decimal degrees are simply a different way to express an angle’s measurement, breaking down each degree into decimal form. This makes calculations involving very specific angles more straightforward and minimizes rounding errors. For instance, an angle of \( 51.228 \) degrees is more precise than stating it as \( 51^\circ \) using traditional degree-minute-second (DMS) notation.When calculating angles like \( \theta \), an inverse trigonometric function's result may be presented in decimal degrees, ensuring high precision. Ensure your calculator is correctly set to degrees mode for accurate results in your trigonometry calculations.