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Magazine Chapter 7: Problem 39
Evaluate. \(\cos ^{-1}\left[\cos \left(-\frac{\pi}{4}\right)\right]\)
Short Answer
Expert verified
The answer is \(\frac{\text{蟺}}{4}\).
Step by step solution
01
Understand the problem
The given problem requires evaluating \(\text{cos}^{-1}\big[\text{cos}(-\frac{\text{蟺}}{4})\big]\). Recognize that \(\text{cos}^{-1}\) is the inverse cosine function and will output an angle whose cosine is the given value.
02
Simplify the inner expression
First, evaluate the inner expression \(\text{cos}(-\frac{\text{蟺}}{4})\). Recall that cosine is an even function, which implies \(\text{cos}(-x) = \text{cos}(x)\). Therefore, \(\text{cos}(-\frac{\text{蟺}}{4}) = \text{cos}(\frac{\text{蟺}}{4})\).
03
Evaluate the cosine value
Next, recall the value of \(\text{cos}(\frac{\text{蟺}}{4})\). By trigonometric knowledge, \(\text{cos}(\frac{\text{蟺}}{4}) = \frac{\text{鈭�2}}{2}\).
04
Apply the inverse cosine function
Now, we need to find an angle \(\theta\) such that \(\text{cos}(\theta) = \frac{\text{鈭�2}}{2}\). In the range \([0, \text{蟺}]\), the value \(\frac{\text{鈭�2}}{2}\) corresponds to \(\theta = \frac{\text{蟺}}{4}\).
05
Write the final answer
Thus, \(\text{cos}^{-1}\big[\text{cos}(-\frac{\text{蟺}}{4})\big] = \frac{\text{蟺}}{4}\).
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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 relates to the angle in a right-angled triangle to the ratio of the length of the adjacent side over the hypotenuse. In mathematical terms, for an angle \(\theta\), it is defined as \(\text{cos}(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\).
Cosine is one of the fundamental trigonometric functions and is commonly denoted as \(\text{cos}(x)\).
Key points about the cosine function:
Cosine is one of the fundamental trigonometric functions and is commonly denoted as \(\text{cos}(x)\).
Key points about the cosine function:
- It is periodic with period \(2\text{蟺}\), meaning \(\text{cos}(x) = \text{cos}(x + 2\text{蟺})\).
- The cosine function ranges from -1 to 1, i.e., \(-1 \leq \text{cos}(x) \leq 1\).
- At specific angles like \(0, \frac{\text{蟺}}{2}, \text{蟺}, \text{etc.}\), cosine values are easily determined.
evaluating trigonometric expressions
Evaluating trigonometric expressions involves determining the values of trigonometric functions for given angles. In our exercise, we evaluated \(\text{cos}(-\frac{\text{蟺}}{4})\).
Here's how to evaluate such expressions step by step:
Here's how to evaluate such expressions step by step:
- Identify the trigonometric function (e.g., sine, cosine, tangent).
- Determine if any properties of the function, such as even or odd functions, can simplify the calculation.
- Use known values or reference angles to find the value. For example, knowing that \(\text{cos}(\frac{\text{蟺}}{4}) = \frac{\text{鈭�2}}{2}\) is handy.
- Apply these known values or identities to solve the expression.
- \(\text{cos}(0) = 1\)
- \(\text{cos}(\frac{\text{蟺}}{2}) = 0\)
- \(\text{cos}(\text{蟺}) = -1\)
even and odd functions
Functions are categorized as even or odd based on their symmetry properties. This concept helps simplify trigonometric expressions.
Knowing whether a function is even or odd helps in quickly evaluating expressions and understanding the function's behavior in different scenarios.
For instance, using the property that cosine is even, we know \(\text{cos}(-\frac{\text{蟺}}{4}) = \text{cos}(\frac{\text{蟺}}{4})\) without further computation.
This inherent symmetry plays a significant role in simplifying trigonometric problems.
- **Even Functions**: For an even function \(f(x)\), \(f(-x) = f(x)\). This means the function's graph is symmetric with respect to the y-axis. The cosine function is an example of an even function. Hence, \(\text{cos}(-x) = \text{cos}(x)\), which was used in our exercise.
- **Odd Functions**: For an odd function \(f(x)\), \(f(-x) = -f(x)\). This means the function's graph is symmetric with respect to the origin. An example is the sine function where \(\text{sin}(-x) = -\text{sin}(x)\).
Knowing whether a function is even or odd helps in quickly evaluating expressions and understanding the function's behavior in different scenarios.
For instance, using the property that cosine is even, we know \(\text{cos}(-\frac{\text{蟺}}{4}) = \text{cos}(\frac{\text{蟺}}{4})\) without further computation.
This inherent symmetry plays a significant role in simplifying trigonometric problems.
