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Chapter 6: Problem 61

Find the function value. Round to four decimal places. $$\cos 40.35^{\circ}$$

Short Answer

Expert verified
0.7615

Step by step solution

01

- Identify the function and angle

The function given is the cosine function, and the angle provided is 40.35 degrees.
02

- Use a calculator

Input the angle into a scientific calculator and use the cosine function to find the value of \( \cos 40.35^{\circ} \).
03

- Calculate the value

Upon calculation, the value of \( \cos 40.35^{\circ} \) is approximately 0.7615 when rounded to four decimal places.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

cosine function
The cosine function is one of the primary trigonometric functions. It relates the angle in a right triangle to the ratio of the length of the adjacent side to the hypotenuse. Mathematically, it is expressed as:
$$ \text{cos}(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} $$
where:
  • \theta is the angle
  • adjacent is the length of the side adjacent to the angle
  • hypotenuse is the length of the longest side of the triangle
The cosine function is also periodic, repeating its values at regular intervals. It is especially important in various fields, from engineering to computer science, for modeling cycles and waves.
angle measurement
Angle measurement is crucial when dealing with trigonometric functions. Angles are typically measured in degrees or radians. In the given exercise, the angle is 40.35 degrees.
Knowing how to convert between degrees and radians can be helpful, as some contexts, like calculus, commonly use radians. The conversion between degrees and radians is given by:
\(\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\text{\text{蟺}}}{180}\)
where 蟺 (pi) is approximately 3.14159.
For most everyday trigonometric problems, degrees are the standard, making them easier to understand and work with.
using a calculator
Using a scientific calculator is essential for finding the values of trigonometric functions like cosine, particularly for non-standard angles like 40.35 degrees. Follow these simple steps to calculate trigonometric function values:
  • Ensure your calculator is in the correct mode (degrees or radians) depending on the given angle measurement.
  • Input the angle value.
  • Press the trigonometric function button (cos for cosine in our example).
  • View and round the result to the required decimal places, in this case to four decimal places.
For instance, to find \(\text{cos}(40.35^{\text{掳}})\), make sure the calculator is in degree mode. Input 40.35 followed by pressing the 'cos' button. The displayed value should be approximately 0.7615 after rounding. This approach ensures accuracy and efficiency when solving trigonometric problems.

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