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Chapter 7: Problem 47

Evaluate. \(\tan \left(\cos ^{-1} \frac{\sqrt{2}}{2}\right)\)

Short Answer

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Step by step solution

01

Understand the Problem

The problem asks to evaluate \(\tan \left(\text{cos}^{-1} \frac{\sqrt{2}}{2}\right)\). This involves finding the tangent of an angle whose cosine is given by \(\frac{\sqrt{2}}{2}\).
02

Identify the Angle

Recall that \(\text{cos}^{-1} \frac{\sqrt{2}}{2}\) represents the angle whose cosine is \(\frac{\sqrt{2}}{2}\). This angle is \(\frac{\pi}{4}\), since \(\text{cos} \frac{\pi}{4} = \frac{\sqrt{2}}{2}\).
03

Compute the Tangent

Now, we need to find the tangent of \(\frac{\pi}{4}\). Recall that \(\tan \frac{\pi}{4} = 1\).
04

Write the Final Answer

Therefore, \(\tan \left(\text{cos}^{-1} \frac{\sqrt{2}}{2}\right) = 1\).

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Key Concepts

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

inverse trigonometric functions
Inverse trigonometric functions help us find the angles when the value of the trigonometric function is known. For example, \(\text{cos}^{-1}\frac{\text{value}}{\text{hypotenuse}}\) tells us the angle whose cosine is the given value. These functions include \(\text{cos}^{-1}\), \(\text{sin}^{-1}\), and \(\text{tan}^{-1}\). They're essential in solving problems where we need to determine an angle from a known ratio.
tangent function
The tangent function, denoted as \(\tan(\theta)\), represents the ratio of the opposite side to the adjacent side in a right triangle. It's one of the primary trigonometric functions. Tangent of \(\theta\) can be defined as:
  • \(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\)
Tangent also relates to sine and cosine with the identity:
  • \(\tan(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)}\)
cosine function
The cosine function, represented as \(\text{cos}(\theta)\), measures the ratio of the adjacent side to the hypotenuse in a right triangle. For any angle \(\theta\), simplifying the related angle evaluations often uses identities such as:
  • \(\text{cos}(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
We also note common angle values:
  • \(\text{cos}(\frac{\text{蟺}}{4}) = \frac{\text{鈭�2}}{2}\)
Identifying these angles is crucial in solving trigonometric problems.
angle evaluation
When evaluating angles, it's essential to connect known values to trigonometric identities. For example, knowing \(\text{cos}^{-1}\) represents an angle whose cosine is a specific value. For \(\text{cos}^{-1}\frac{\text{鈭�2}}{2}\), the angle is \(\frac{\text{蟺}}{4}\). Compute tangent by using \(\tan(\frac{\text{蟺}}{4})\). These steps clarify trigonometric functions effectively.

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Most popular questions from this chapter

Find the following. \(\tan \left(\frac{1}{2} \sin ^{-1} \frac{1}{2}\right)\)

Electrical Theory. In electrical theory, the following equations occur: $$E_{1}=\sqrt{2} E_{t} \cos \left(\theta+\frac{\pi}{P}\right)$$ and $$E_{2}=\sqrt{2} E_{t} \cos \left(\theta-\frac{\pi}{P}\right)$$. Assuming that these equations hold, show that $$\frac{E_{1}+E_{2}}{2}=\sqrt{2} E_{t} \cos \theta \cos \frac{\pi}{P}$$ and $$\frac{E_{1}-E_{2}}{2}=-\sqrt{2} E_{t} \sin \theta \sin \frac{\pi}{P}$$.

Simplify. $$\cos (u+v) \cos v+\sin (u+v) \sin v$$

In the theory of alternating current, the following equation occurs: $$R=\frac{1}{\omega C(\tan \theta+\tan \phi)}$$ Show that this equation is equivalent to $$R=\frac{\cos \theta \cos \phi}{\omega C \sin (\theta+\phi)}$$.

Evaluate. \(\tan ^{-1}\left[\tan \left(-\frac{3 \pi}{4}\right)\right]\)
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