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Chapter 1: Problem 212

Evaluate the functions. Give the exact value. \(\cos \left(\tan ^{-1}(\sqrt{3})\right)\)

Short Answer

Expert verified
The exact value is \( \frac{1}{2} \).

Step by step solution

01

Understand the Inverse Tangent Function

The expression involves \( \tan^{-1}(\sqrt{3}) \). This function asks for the angle \( \theta \) such that \( \tan(\theta) = \sqrt{3} \).
02

Identify the Angle

From trigonometric values, we know that \( \tan(\pi/3) = \sqrt{3} \). Thus, \( \tan^{-1}(\sqrt{3}) = \pi/3 \).
03

Substitute and Simplify

Now, substitute the angle into the cosine function: \( \cos(\tan^{-1}(\sqrt{3})) = \cos(\pi/3) \).
04

Evaluate the Cosine Function

Recall that \( \cos(\pi/3) = 1/2 \). Thus, \( \cos(\tan^{-1}(\sqrt{3})) = 1/2 \).

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Key Concepts

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

Understanding Inverse Tangent
The inverse tangent function, often denoted as \( \tan^{-1}(x) \) or "arctan," helps us find an angle \( \theta \) when the tangent of that angle is known. This function essentially reverses the tangent function. In our exercise, we have \( \tan^{-1}(\sqrt{3}) \). This means we're looking for an angle where its tangent value is \( \sqrt{3} \). This will often relate back to what we call "special angles," which are angles whose trigonometric values are well-known.
  • In trigonometry, the tangent of \( \pi/3 \) is \( \sqrt{3} \).
  • Thus, \( \tan^{-1}(\sqrt{3}) \) is identical to \( \pi/3 \), meaning the angle \( \theta \) is \( \pi/3 \).
Understanding these relationships helps us make connections between angles and their trigonometric functions effectively.
The Role of the Cosine Function
The cosine function is crucial for evaluating the trigonometric expression \( \cos(\tan^{-1}(\sqrt{3})) \). The cosine of an angle, represented as \( \cos(\theta) \), relates to the adjacent side over the hypotenuse in a right triangle. Once we've identified \( \theta \) from the \( \tan^{-1} \) function, we can use this angle within the cosine function.
  • From earlier, we found that \( \tan^{-1}(\sqrt{3}) = \pi/3 \).
  • The cosine of \( \pi/3 \) is a well-known trigonometric value, which is \( \frac{1}{2} \).
Hence, upon substituting it into the cosine function, we find \( \cos(\tan^{-1}(\sqrt{3})) = \cos(\pi/3) = \frac{1}{2} \). Recognizing and memorizing these values can simplify trigonometry problems considerably.
Common Trigonometric Values and Angles
Understanding the specific values of trigonometric functions for certain angles is very helpful in trigonometry. These special angles, such as \( \pi/6 \), \( \pi/4 \), and \( \pi/3 \), have trigonometric values that make calculations straightforward. For example,
  • \( \sin(\pi/3) = \frac{\sqrt{3}}{2} \)
  • \( \cos(\pi/3) = \frac{1}{2} \)
  • \( \tan(\pi/3) = \sqrt{3} \)
These values are not arbitrary; they come from the properties of equilateral and special right triangles.
Remembering these trigonometric values allows you to solve problems more easily and recognize patterns quickly. They serve as a key foundation in understanding more complex trigonometric identities and equations.

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Most popular questions from this chapter

ITI The output (as a percent of total capacity) of nuclear power plants in the United States can be modeled by the function \(P(t)=1.8576 t+68.052,\) where \(t\) is time in years and \(t=0\) corresponds to the beginning of 2000 . Use the model to predict the percentage output in 2015 .

For the following exercises, use the change-of-base formula and either base 10 or base \(e\) to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places. $$ \log _{6} 103 $$

For the following exercises, write the equation in equivalent logarithmic form. $$ b^{3}=45 $$

For the following exercises, solve the logarithmic equation exactly, if possible. $$ \log _{4}(x+2)-\log _{4}(x-1)=0 $$

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