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Chapter 5: Problem 1

For Exercises 1-25, find the exact value of the given expression in radians. \(\tan ^{-1} 1\)

Short Answer

Expert verified
\(\tan^{-1}(1) = \frac{\

Step by step solution

01

Understanding Inverse Tangent

The function \(\tan^{-1}(x)\) gives the angle whose tangent is \(x\). In other words, \(\tan^{-1}(1)\) is the angle whose tangent is 1.
02

Recall the Tangent Values of Common Angles

Recall that \(\tan(\frac{\pi}{4}) = 1\). This means that \(\frac{\pi}{4}\) radians is the angle whose tangent is 1.
03

Find the Exact Value

Based on the recall from common angles, the exact value of \(\tan^{-1}(1)\) is \(\frac{\pi}{4}\) radians.

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Key Concepts

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

inverse tangent
The inverse tangent, also written as \(\tan^{-1}(x)\) or arctan(x), is a fundamental concept in trigonometry. It answers a specific question: 'For which angle \(\theta\) does the tangent equal \(x\)?
exact value
Finding the exact value of inverse trigonometric functions means identifying the precise angle, often in terms of radians or degrees, without any approximation. For example, the exact value of \(\tan^{-1}(1)\) is \ \frac{\pi}{4}\ radians because \(\tan(\frac{\pi}{4}) = 1\). Using known relations between angles and trigonometric functions helps in obtaining the exact answer quickly.
radians
Radians are a unit of angular measurement. Unlike degrees, which divide a circle into 360 parts, radians measure angles in terms of the radius of the circle. One complete revolution around a circle is \(2\pi\) radians. For instance, \(\frac{\pi}{4}\) radians is equivalent to 45 degrees, providing a more natural way to describe angles, especially in trigonometry. For trigonometric problems, angles are often preferred to be in radians.

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Most popular questions from this chapter

Prove the given identity. \(\tan ^{-1} x+\tan ^{-1} \frac{1}{x}=\frac{\pi}{2} \quad\) for \(x>0\)

Find the exact value of the given expression in radians. \(\tan ^{-1} \frac{3}{5}+\cot ^{-1} \frac{3}{5}\)

Prove the given identity. \(\sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}\)

Find the exact value of the given expression in radians. \(\tan ^{-1}\left(\tan \left(-\frac{5 \pi}{6}\right)\right)\)

For any point \((x, y)\) on the unit circle and any angle \(\alpha\), show that the point \(R_{\alpha}(x, y)\) defined by \(R_{\alpha}(x, y)=(x \cos \alpha-y \sin \alpha, x \sin \alpha+y \cos \alpha)\) is also on the unit circle. What is the geometric interpretation of \(R_{\alpha}(x, y)\) ? Also, show that \(R_{-\alpha}\left(R_{\alpha}(x, y)\right)=(x, y)\) and \(R_{\beta}\left(R_{\alpha}(x, y)\right)=R_{\alpha+\beta}(x, y)\).
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