Problem 227
Question
Use a calculator to evaluate \(\tan ^{-1}(\tan (2.1))\) and \(\cos ^{-1}(\cos (2.1)) .\) Explain the results of each.
Step-by-Step Solution
Verified Answer
\( \tan^{-1}(\tan(2.1)) \approx 2.1 \); \( \cos^{-1}(\cos(2.1)) \approx 2.638 \).
1Step 1: Understanding Inverse Functions
The notation \( \tan^{-1} \) and \( \cos^{-1} \) represent the inverse functions of tangent and cosine, respectively. Evaluating these requires considering the principal values, which are the standard ranges associated with these inverse functions.
2Step 2: Calculate \( \tan(2.1) \)
First, use a calculator to find \( \tan(2.1) \). Enter 2.1 into the calculator in radians mode (since no unit is specified, radians are typically used) and press the \( \tan \) function. This yields a value of approximately 0.969.
3Step 3: Evaluate \( \tan^{-1}(0.969) \)
Find the angle whose tangent is 0.969 using a calculator by entering 0.969 and pressing the \( \tan^{-1} \) button. This results in a value close to 2.1 because the input angle is within the principal range \( (-\pi/2, \pi/2) \) for \( \tan^{-1} \).
4Step 4: Calculate \( \cos(2.1) \)
Next, use a calculator to find \( \cos(2.1) \). Ensure the calculator is in radians mode, enter 2.1, and press the \( \cos \) function. This results in a value of approximately -0.504.
5Step 5: Evaluate \( \cos^{-1}(-0.504) \)
Find the angle whose cosine is -0.504 using a calculator by entering -0.504 and pressing the \( \cos^{-1} \) button. This returns an angle not equal to 2.1, instead it's approximately 2.638 which is in the range \( [0, \pi] \) for \( \cos^{-1} \).
6Step 6: Analyze and Explain the Results
For \( \tan^{-1}(\tan(2.1)) \), the result is approximately 2.1 because 2.1 falls within the principal range of \( \tan^{-1} \). However, for \( \cos^{-1}(\cos(2.1)) \), the result is not 2.1, because the input 2.1 is outside its principal range, and the cosine function is symmetric around the y-axis.
Key Concepts
Principal ValueTangent FunctionCosine Function
Principal Value
When you encounter inverse functions like \( \tan^{-1} \) or \( \cos^{-1} \), it's important to understand the concept of principal value. Principal values are specific ranges in which the inverse trigonometric functions output their values. These ranges ensure that each output value is unique for a given input.
For \( \tan^{-1} \), the principal value lies in the range \( (-\pi/2, \pi/2) \). This range ensures any value you input for \( \tan^{-1} \) results in an output exclusively between these limits. Similarly, the principal range for \( \cos^{-1} \) is \( [0, \pi] \).
Understanding these principal values is essential because they determine how the inverse functions interpret values fed into them. If a value is outside these ranges, the function automatically maps it to a value within this principal range.
For \( \tan^{-1} \), the principal value lies in the range \( (-\pi/2, \pi/2) \). This range ensures any value you input for \( \tan^{-1} \) results in an output exclusively between these limits. Similarly, the principal range for \( \cos^{-1} \) is \( [0, \pi] \).
Understanding these principal values is essential because they determine how the inverse functions interpret values fed into them. If a value is outside these ranges, the function automatically maps it to a value within this principal range.
Tangent Function
The tangent function, denoted as \( \tan(x) \), is a fundamental trigonometric function which compares the length of the opposite side to the adjacent side in a right-angled triangle. It's important to remember that the tangent function has a period of \( \pi \), meaning its pattern repeats every \( \pi \) radians.
An interesting aspect of the tangent function is that it is not defined for all angles. There are certain points, such as \( \frac{\pi}{2} + k\pi \) (where \( k \) is any integer), where \( \tan(x) \) becomes undefined. This happens because the function approaches infinity, creating vertical asymptotes in its graph.
When using the inverse \( \tan^{-1} \) function, it aims to return an angle whose tangent equals the specified value. Because the range is restricted to \( (-\pi/2, \pi/2) \), as long as the tangent value falls within this range, the solution will be unique. Thus, the principal value of the tangent function ensures continuity and predictability in its calculations.
An interesting aspect of the tangent function is that it is not defined for all angles. There are certain points, such as \( \frac{\pi}{2} + k\pi \) (where \( k \) is any integer), where \( \tan(x) \) becomes undefined. This happens because the function approaches infinity, creating vertical asymptotes in its graph.
When using the inverse \( \tan^{-1} \) function, it aims to return an angle whose tangent equals the specified value. Because the range is restricted to \( (-\pi/2, \pi/2) \), as long as the tangent value falls within this range, the solution will be unique. Thus, the principal value of the tangent function ensures continuity and predictability in its calculations.
Cosine Function
The cosine function, expressed as \( \cos(x) \), measures the adjacent side's length relative to the hypotenuse in a right-angled triangle. Unlike the tangent function, cosine is defined for any real value because it oscillates between -1 and 1 and has a period of \( 2\pi \).
A significant feature of the cosine function is its symmetry about the y-axis, which implies that it has the property \( \cos(x) = \cos(-x) \). This symmetry is why, when using the inverse \( \cos^{-1} \) function, the output might not always be what you initially expect.
For \( \cos^{-1} \), the principal range \( [0, \pi] \) ensures it consistently outputs angles in this interval. If an angle is beyond this range, the function will adjust to find an equivalent angle within the principal range. This behavior is why \( \cos^{-1}(\cos(2.1)) \) gives a result of about 2.638 rather than returning 2.1 directly. This adjustment is necessary to maintain the function's consistency within its defined principal range.
A significant feature of the cosine function is its symmetry about the y-axis, which implies that it has the property \( \cos(x) = \cos(-x) \). This symmetry is why, when using the inverse \( \cos^{-1} \) function, the output might not always be what you initially expect.
For \( \cos^{-1} \), the principal range \( [0, \pi] \) ensures it consistently outputs angles in this interval. If an angle is beyond this range, the function will adjust to find an equivalent angle within the principal range. This behavior is why \( \cos^{-1}(\cos(2.1)) \) gives a result of about 2.638 rather than returning 2.1 directly. This adjustment is necessary to maintain the function's consistency within its defined principal range.
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