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Chapter 6: Problem 86

Find the function value. Round to four decimal places. $$\cos 205.5^{\circ}$$

Short Answer

Expert verified
\[ \text{cos}(205.5^{\text{掳}}) \approx -0.9135 \]

Step by step solution

01

Convert degrees to radians

To find the cosine value, first convert the angle from degrees to radians. Use the formula \(\text{radians} = \text{degrees} \times \frac{\text{蟺}}{180}\). So, \[ \text{radians} = 205.5^{\text{掳}} \times \frac{\text{蟺}}{180} = 3.5864 \text{ radians} \]
02

Use cosine function

With the angle in radians, apply the cosine function: \(\text{cos}(3.5864)\).
03

Round the result

Calculate the cosine value using a calculator or cosine table: \[ \text{cos}(205.5^{\text{掳}}) \approx -0.9135 \]

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Key Concepts

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

Degree to Radian Conversion
Before finding the value of the cosine function, we need to convert our angle from degrees to radians. This step is crucial because most trigonometric functions in higher-level math are defined using radians. The standard formula to convert degrees to radians is:

  • \(\text{radians} = \text{degrees} \times \frac{\text{蟺}}{180}\)
Example

For our given exercise:
We need to convert 205.5 degrees to radians.
Using the formula:

\(\text{radians} = 205.5 \times \frac{\text{蟺}}{180} = 3.5864 \)
Therefore, 205.5 degrees is equivalent to 3.5864 radians.
Trigonometric Functions
Trigonometric functions are fundamental in studying angles and relationships in right-angled triangles. The most common trigonometric functions are sine (\(\text{sin}\)), cosine (\(\text{cos}\)), and tangent (\(\text{tan}\)). In this exercise, we focus on the cosine function.

  • The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse.
However, when dealing with angles like 205.5 degrees, which are not part of a right triangle, we can still find the cosine value using a calculator. Once we have our angle in radians:
We simply apply the cosine function:
\(\text{cos}(3.5864)\)

Using a scientific calculator or a software tool, we find:
\(\text{cos}(205.5^{\text{掳}}) \text{ or } \text{cos}(3.5864) \text{ which is approximately } -0.9135\).
Rounding
Rounding numbers is an essential skill in math, especially when dealing with long decimal results. Often, we are asked to round to a certain number of decimal places for simplicity and clarity.

In our exercise, we've been instructed to round the final cosine value to four decimal places,
Here's a quick guide:
  • Identify the digit in the place you're rounding to (the fourth decimal place).
  • Look at the next digit (the fifth decimal place).
  • If the fifth digit is 5 or more, round up the fourth digit by 1.
  • If the fifth digit is less than 5, leave the fourth digit as is.
For our result:
\(\text{cos}(205.5^{\text{掳}}) \text{ or } \text{cos}(3.5864) \text{ is approximately } -0.9135\). This means:
The fifth digit (beyond 4 decimal places) was small enough to leave -0.9135 unchanged at four decimal places.
Thus, the rounded value to four decimal places is -0.9135.

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Most popular questions from this chapter

Find the reference angle and the exact function value if they exist. $$\sin \left(-135^{\circ}\right)$$

Use a graphing calculator to graph the function. $$y=-(\cos x-x)$$

Find the reference angle and the exact function value if they exist. $$\cot \left(-90^{\circ}\right)$$

Find \(f \circ g\) and \(g \circ f,\) where \(f(x)=x^{2}+2 x\) and \(g(x)=\cos x\)

Use a graphing calculator to graph each of the following on the given interval and approximate the zeros. $$f(x)=x^{3} \sin x ;[-5,5]$$
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