Problem 16
Question
In Problems 11-18, use a calculator to approximate each value. \(\tan ^{-1}(-60.11)\)
Step-by-Step Solution
Verified Answer
\(\tan^{-1}(-60.11)\) is approximately -89.0484 degrees.
1Step 1: Understand the problem
We need to find the angle whose tangent is -60.11. This problem involves finding the inverse tangent (also called arctangent) of -60.11, expressed mathematically as \(\tan^{-1}(-60.11)\).
2Step 2: Use a calculator
To solve for \(\tan^{-1}(-60.11)\), use a calculator with inverse trigonometric functions. Ensure the calculator is set to the correct mode (degrees or radians, depending on your requirement). Enter -60.11 and use the inverse tangent function to get the result.
3Step 3: Interpret the calculator result
When you compute \(\tan^{-1}(-60.11)\) using the calculator, you might get a result approximately equal to -89.0484 degrees or in radians, depending on your calculator mode. Arctangent results are typically given in the range from -\(\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians or -90 to 90 degrees.
Key Concepts
Inverse Trigonometric FunctionsArctangentCalculators in MathematicsAngle Measurement
Inverse Trigonometric Functions
Inverse trigonometric functions are a fascinating aspect of trigonometry. They allow us to find angles when given trigonometric ratios, essentially working backwards from regular trigonometric functions. For example, if we know the tangent of an angle is 0.5, we can use the inverse tangent, or arctangent, to find the angle itself.
These functions are inverse to the basic trigonometric functions:
These functions are inverse to the basic trigonometric functions:
- The inverse of sine is arcsine (\( \sin^{-1}(x) \)
- The inverse of cosine is arccosine (\( \cos^{-1}(x) \)
- The inverse of tangent is arctangent (\( \tan^{-1}(x) \)
- Arcsine and arccosine outputs are between -90 and 90 degrees.
- Arctangent outputs are between -90 and 90 degrees.
Arctangent
Arctangent is the specific inverse trigonometric function that we use when dealing with the tangent value of an angle. It's notated as \( \tan^{-1}(x) \) or arctan. Essentially, when you see \( \tan^{-1}(x) \), you're looking for the angle whose tangent value is \(x\).
One unique feature of the arctangent function is its range, which is restricted from -\( \frac{\pi}{2} \) radians to \( \frac{\pi}{2} \) radians (or -90 to 90 degrees). This restriction ensures that every input produces a specific angle output. So, whether you're dealing with a large positive or negative number, arctan finds an angle within these bounds, making it a very practical tool in trigonometry.
In practical scenarios, you might see questions like: "What angle has a tangent of -60.11?" Using \( \tan^{-1}(-60.11) \) with a calculator gives you the corresponding angle measurement. This is particularly useful in various fields such as physics, engineering, and computer science, where understandings of angles and proportions are crucial.
One unique feature of the arctangent function is its range, which is restricted from -\( \frac{\pi}{2} \) radians to \( \frac{\pi}{2} \) radians (or -90 to 90 degrees). This restriction ensures that every input produces a specific angle output. So, whether you're dealing with a large positive or negative number, arctan finds an angle within these bounds, making it a very practical tool in trigonometry.
In practical scenarios, you might see questions like: "What angle has a tangent of -60.11?" Using \( \tan^{-1}(-60.11) \) with a calculator gives you the corresponding angle measurement. This is particularly useful in various fields such as physics, engineering, and computer science, where understandings of angles and proportions are crucial.
Calculators in Mathematics
Calculators are indispensable tools in mathematics, especially when dealing with complex functions like trigonometric inverses. Most scientific calculators come equipped with functions for sine, cosine, tangent, and their inverses (arcsine, arccosine, and arctangent).
When you use a calculator to find \( \tan^{-1}(x) \), it's crucial to ensure that the calculator is set to the correct unit of measurement — degrees or radians — depending on your context. By inputting the value for which you wish to find the arctangent, the calculator will quickly output the precise angle, sparing you the intricacy of manual computations.
While using a calculator is straightforward, interpreting the results correctly is as important. Calculators can return results in either degrees or radians, and knowing what your calculator is set for will help you avoid misinterpretation of the angles you are working with.
When you use a calculator to find \( \tan^{-1}(x) \), it's crucial to ensure that the calculator is set to the correct unit of measurement — degrees or radians — depending on your context. By inputting the value for which you wish to find the arctangent, the calculator will quickly output the precise angle, sparing you the intricacy of manual computations.
While using a calculator is straightforward, interpreting the results correctly is as important. Calculators can return results in either degrees or radians, and knowing what your calculator is set for will help you avoid misinterpretation of the angles you are working with.
Angle Measurement
Understanding angle measurement is key, as angles can be measured in two primary ways: degrees and radians. Degrees are more commonly used in everyday applications, while radians are frequently used in mathematical contexts, especially concerning calculus and trigonometry.
- Degrees: A full circle is 360 degrees.
- Radians: A full circle is \( 2 \pi \) radians.
- To convert degrees to radians, multiply by \( \frac{\pi}{180} \).
- To convert radians to degrees, multiply by \( \frac{180}{\pi} \).
Other exercises in this chapter
Problem 16
Sketch the graph of \(g(x)=|x+3|-4\) by first sketching \(h(x)=|x|\) and then translating.
View solution Problem 16
In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(f(x)=3 x\)
View solution Problem 16
Express the solution set of the given inequality in interval notation and sketch its graph. $$ \frac{3 x-2}{x-1} \geq 0 $$
View solution Problem 16
Simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. \((\sqrt{5}-\sqrt{3})^{2}\)
View solution