Problem 47 - Algebra 2 and Trigonometry

Question

In \(39-50,\) find the smallest positive value of \(\theta\) to the nearest degree. $$ \sin \theta=0.5446 $$

Step-by-Step Solution

Answer

The smallest positive value of \( \theta \) is \( 33 \) degrees.

Step 1: Understanding the Problem

The problem requires us to find the smallest positive angle \( \theta \) such that \( \sin \theta = 0.5446 \). We are asked to find this angle to the nearest degree.

Step 2: Using the Inverse Sine Function

To find \( \theta \), we use the inverse sine function (or arcsin). This function helps us find the angle whose sine value is \( 0.5446 \). We calculate \( \theta = \sin^{-1}(0.5446) \).

Step 3: Calculating \( \theta \)

Using a calculator, we find \( \theta \approx 32.9331 \) degrees.

Step 4: Rounding to the Nearest Degree

We are asked to round \( \theta \) to the nearest whole number. So, we round \( 32.9331 \) to \( 33 \) degrees.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in trigonometry. They help us find the angles associated with specific trigonometric values. These functions are the inverse operations of the primary trigonometric functions: sine, cosine, and tangent.

When you have a value of a trigonometric function, say sine, and you need to find the corresponding angle, you would use the inverse. For instance, the inverse sine function (also known as arcsine) is written as \( \sin^{-1}(x) \) and finds the angle whose sine is \( x \).

Here are some key points about inverse trigonometric functions:

• They provide angles within specific ranges that are considered principal values. For sine, this range is \([-\frac{\pi}{2}, \frac{\pi}{2}]\) or \([-90^{\circ}, 90^{\circ}]\).
• They are crucial for solving equations where the angle needs to be determined from a trigonometric ratio.

By understanding inverse functions, you can convert a trigonometric value back into an angle, which is very handy in many mathematical applications, including this exercise.

Angle Calculation

Angle calculation is a vital skill in trigonometry, as it involves determining the exact measure of an angle based on a given trigonometric value.

To perform an angle calculation using the inverse trigonometric functions:

• Identify the trigonometric function and its given value.
• Use the appropriate inverse function to find the angle. For example, with \( \sin \theta = 0.5446\), use the inverse sine function.
• Interpret and adjust the results to the required unit (degrees or radians) if necessary. In this case, we aim for degrees.

After computing the inverse sine of \(0.5446\), you might get a decimal degree result like \(32.9331\).

To finalize, you often need to round off your answer, just like rounding \(32.9331\) to \(33\) degrees.

It is helpful to remember that calculators typically give results in default units, so ensure your calculator is set to the desired unit before starting these calculations.

Sine Function

The sine function is one of the three primary trigonometric functions used to relate the angles and sides of right triangles.

It’s defined as the ratio of the length of the opposite side of an angle to the hypotenuse in such a triangle. For an angle \( \theta \):

• \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \)

The sine function is periodic with a fundamental period of \(360^{\circ}\) or \(2\pi\) radians, meaning it repeats its values in these intervals.

Some key points about the sine function include:

• It has a range of \([-1, 1]\), as the output is always a fraction of the hypotenuse length.
• It is useful in modeling cyclical phenomena like waves.
• The curve of \( \sin \theta \) starts at 0, rises to 1, decreases back to 0, dips down to -1, then returns back to 0 across the span of \(360^{\circ}\).

When solving trigonometric equations like \( \sin \theta = 0.5446 \), understanding this function allows us to locate the angle from within its cyclical pattern that satisfies the equation.