Problem 21 - 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

\(\theta = 46\) degrees.

Step 1: Understanding the Problem

We need to find the angle \(\theta\) such that \(\theta = \arcsin(0.72)\). The arcsin function, also known as the inverse sine function, returns the angle whose sine is the given number, in this case, 0.72.

Step 2: Using a Calculator to Find Theta

To find \(\theta\), enter 0.72 into the calculator and apply the arcsin or inverse sine function (often denoted as \(\sin^{-1}\) or \(\arcsin\)). Make sure the calculator is set to 'degree' mode to get the answer in degrees.

Step 3: Reading and Rounding the Result

After calculating, you will find that \(\theta \approx 46.05\) degrees. As the problem asks for the nearest degree, round this value to the nearest whole number.

Step 4: Concluding with the Final Answer

After rounding, we find that \(\theta = 46\) degrees.

Key Concepts

Trigonometric functions

Trigonometric functions are fundamental to understanding angles and relationships in mathematics. They include sin, cos, and tan functions. These functions help in relating the angles of a triangle to its side lengths. A typical use of these functions is when dealing with right-angled triangles. Here, the sine function helps us find the ratio of the length of the opposite side to the hypotenuse of a triangle. When you're using a calculator to find sine, you are dealing with ratios of these sides in a particular triangle. Understanding how these functions work is crucial for solving problems related to angles.

• They translate angles into usable numeric ratios.
• They are used extensively in physics, engineering, and navigation.

The inverse of these functions allows us to work backwards, helping us calculate angles when we already know the side ratios.

Inverse functions

Inverse functions are a key mathematical concept allowing you to reverse the effect of the original function. These are especially important in trigonometry, where they help find angles from the given ratios of sides. For our exercise, the inverse sine function, denoted as arcsin or \(\sin^{-1}\), is used to find the angle whose sine is a given number, such as 0.72. It's like asking the question "what angle would give us this sine value?"Inverse trigonometric functions turn a simple ratio back into a more complex angle measure. This is especially useful in solving equations where the outcome is known, but the input (angle) isn't.

• Inverse functions are used to determine angle measures.
• These functions help convert a linear number into an angular measure.

Using these functions requires a calculator set in the correct mode to ensure the output is in degrees or radians, as desired.

Angle calculation

Calculating angles using trigonometric functions and their inverses is a common task in geometry and trigonometry. When you are given a ratio, like \(\sin(\theta) = 0.72\), and need to find the angle \(\theta\), the arcsin function comes into play.To determine the angle in this exercise:

• Press the inverse sine key (often labeled as \(\sin^{-1}\) or \(\arcsin\)) on your calculator.
• Enter the number 0.72.
• Ensure the calculator is configured to 'degree' mode.

This will return the angle measurement. In this case, the calculator gives you 46.05 degrees. This process is a straightforward application of trigonometric principles, enabling a numerical relationship to be expressed as an angular measure.

Rounding numbers

Rounding numbers is a basic yet important mathematical skill, especially when dealing with calculations that require a specific precision level. In our exercise, the problem asks to round the angle to the nearest degree. Understanding when and how much to round is key here. When you round 46.05 degrees, you focus on the number after the decimal:

• If it is 5 or more, you round up.
• If it is less than 5, you round down.

Therefore, since 46.05 has 0.05 after the decimal, you round it to 46 degrees. Rounding helps simplify numbers, making them easier to interpret, especially when representing exact values is unnecessary.