Problem 36 - Precalculus Essentials

Question

$$\text {use a calculator to find the value of the acute}\text { angle } \theta \text { to the nearest degree.}$$ $$\cos \theta=0.8771$$

Step-by-Step Solution

Answer

The acute angle \( \theta \) is approximately 29 degrees

Step 1: Identify the mathematical function

From the given information, it is stated that \( \cos \theta = 0.8771 \). The task is to find the angle \( \theta \). Hence, we need to use the inverse cosine function, also known as arccos.

Step 2: Use the inverse cosine function

Applying the inverse cosine function to both sides of the equation yields \( \theta = \cos^{-1}(0.8771) \). Now, calculate this on your calculator.

Step 3: Convert to degrees

As the calculator results might be in radians, ensure to convert the result to degrees if necessary because the problem requires the answer in degrees. The conversion from radian to degrees is done by multiplying the radian value with \( \frac{180}{\pi} \). Therefore, apply this conversion if required and round off to the nearest degree.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are special mathematical functions that allow us to find angles based on the values of trigonometric ratios. These functions are essentially the "reverse" of regular trigonometric functions like sine, cosine, and tangent. For cosine, the inverse function is known as arccosine, often denoted as \( \cos^{-1} \).

• The inverse cosine function, \( \cos^{-1}(x) \), returns the angle \( \theta \) for which the cosine of that angle equals \( x \).
• This function is especially useful when you have a cosine value and need to determine the angle.

In the context of our problem, we use the inverse cosine function to find the angle \( \theta \) when given \( \cos \theta = 0.8771 \). Using a calculator, input \( \cos^{-1}(0.8771) \) to get the result.

Cosine Function

The cosine function is one of the fundamental trigonometric functions, representing the ratio of the adjacent side to the hypotenuse in a right triangle. It is denoted by \( \cos(\theta) \). This function helps relate an angle in a triangle to the lengths of its sides.

• For an acute angle \( \theta \) in a right triangle, \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \).
• The range of the cosine function for angles between 0 and 90 degrees is from 1 (cos(0)) to 0 (cos(90)).

In our problem, the cosine value of an unknown angle is given as 0.8771. To determine this angle \( \theta \), we must use the inverse, \( \cos^{-1} \), function.

Angle Measurement

Angle measurement is a way to determine the size of an angle in a geometric space. Angles can be measured in different units, such as degrees or radians. Understanding the unit of measurement is crucial when interpreting trigonometric values.

• Degrees are part of a circle divided into 360 parts. \( \theta \) is typically measured in degrees when dealing with many real-world applications.
• Radians are based on the radius of a circle and are often used in higher mathematics and theoretical studies. A full circle is \( 2\pi \) radians.

In this problem, knowing whether the final answer is required in degrees or radians is essential. Knowing that the problem specifically asks for degrees, we need to ensure any angle result given in radians is properly converted.

Radians to Degrees Conversion

Conversion between radians and degrees is often necessary in trigonometry. Since some calculators provide angles in radians by default, understanding how to convert these to degrees is essential. The conversion is based on the formula:
\[\text{degrees} = \text{radians} \times \frac{180}{\pi}\]

• Multiply the radian measure by \( \frac{180}{\pi} \) to convert it to degrees.
• This conversion is vital when specific problems, including our example, require results in degrees.

In our example, after finding the angle using the inverse cosine function, convert it with this formula if needed to report the angle in degrees, ensuring it meets the problem's requirements.