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Chapter 5: Problem 2

Evaluate \(\tan \left(\tan ^{-1} 5\right)\).

Short Answer

Expert verified
In conclusion, we find that \(\tan(\tan^{-1}(5)) = 5\).

Step by step solution

01

Find the angle whose tangent is 5

First, we need to find the angle (in radians or degrees) whose tangent is 5. To do this, we will use the inverse tangent function \(\tan^{-1}(x)\): \(\theta = \tan^{-1}(5)\)
02

Evaluate the tangent of the angle found in step 1

Now that we have the angle, we can find the tangent of it. Since the angle is the output of the inverse tangent function of 5 (from the previous step), the tangent of this angle must be equal to 5: \(\tan(\theta) = \tan(\tan^{-1}(5)) = 5\)
03

Write the final answer

In conclusion, we find that \(\tan(\tan^{-1}(5)) = 5\).

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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 are used to find angles when we know the value of a trigonometric function. They are the reverse of the usual trigonometric functions. For example, if we know \( an(\theta) = x\), then \( an^{-1}(x) = \theta\).

These functions are incredibly useful in situations like calculating angles in right triangles or converting between degrees and radians.
  • \(\tan^{-1}(x)\) specifically gives us the angle whose tangent is \(x\).
  • This angle is always within the range \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians (or -90掳 to 90掳).
This property of returning angles within a certain range is crucial in maintaining consistency across calculations.
Tangent Function
The tangent function, denoted by \(\tan\), is one of the primary trigonometric functions alongside sine and cosine. It relates an angle in a right triangle to the ratio of the opposite side over the adjacent side.

If you're on a coordinate plane, \(\tan(\theta)\) is defined as \(\frac{y}{x}\), where \(y\) is the vertical coordinate and \(x\) is the horizontal coordinate.
  • \(\tan(\theta)\) is periodic with a period of \(\pi\) radians, meaning it repeats its values every \(\pi\) radians.
  • The tangent function can take any real number value, from \(-\infty\) to \(\infty\).
Knowing the tangent's behavior helps in understanding how angles map to real number ratios.
Angle Evaluation
Angle evaluation refers to the process of determining the measure of an angle based on trigonometric values. This process often involves using inverse trigonometric functions, like \(\tan^{-1}\), to find the angle.In our example, we evaluated \(\tan(\tan^{-1}(5))\). Here's how it works:
  • First, find the angle whose tangent is 5, using \(\tan^{-1}(5)\).
  • Then, evaluate the tangent of that angle.
By the definition of inverse functions, \(\tan(\tan^{-1}(x)) = x\). This property ensures that once we've determined the angle, computing the tangent brings us back to our original input value, showcasing the inverse relationship effectively.

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Most popular questions from this chapter

Find the smallest positive number \(\theta\) such that \(e^{\tan \theta}=15 .\)

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Show that \(\sin \frac{\pi}{18}\) is a zero of the polynomial \(8 x^{3}-6 x+1\) [Hint: Use the identity from the previous problem.]

Evaluate the indicated expressions assuming that $$ \begin{array}{ll} \cos x=\frac{1}{3} & \text { and } \sin y=\frac{1}{4} \\ \sin u=\frac{2}{3} & \text { and } \cos v=\frac{1}{5} \end{array} $$ Assume also that \(x\) and \(u\) are in the interval \(\left(0, \frac{\pi}{2}\right),\) that \(y\) is in the interval \(\left(\frac{\pi}{2}, \pi\right),\) and that \(v\) is in the interval \(\left(-\frac{\pi}{2}, 0\right) .\) $$\cos (x+y)$$

Show that \(\cos 20^{\circ}\) is a zero of the polynomial \(8 x^{3}-6 x-1\) [Hint: Set \(\theta=20^{\circ}\) in the identity from the previous problem.]
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