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Chapter 6: Problem 10

Evaluate the following expressions, giving the answer in radians. $$ \tan ^{-1}(\sqrt{3}) $$

Short Answer

Expert verified
\( \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \).

Step by step solution

01

Understand the Expression

We need to evaluate the inverse tangent function, also known as the arctangent, of \( \sqrt{3} \). That is, find the angle whose tangent is \( \sqrt{3} \).
02

Recall Tangent Values

Recall from trigonometric identities that the tangent of \( \frac{\pi}{3} \) (60 degrees) is \( \sqrt{3} \). This means \( \tan(\frac{\pi}{3}) = \sqrt{3} \).
03

Apply the Inverse Tangent

Since the tangent of \( \frac{\pi}{3} \) is \( \sqrt{3} \), the inverse tangent of \( \sqrt{3} \) is \( \frac{\pi}{3} \). Thus, \( \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \).

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Key Concepts

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

Arctangent
The arctangent, often written as \( \tan^{-1}(x) \), is an important concept in trigonometry. It is the inverse function of the tangent. This means it helps us find the angle when we know the tangent value. Think of it as the opposite of the tangent function. If tangent tells us the ratio from an angle, arctangent gives us back the angle from the ratio. This is why we use arctangent to find angles based on known tangent values. Here's a simple breakdown:
  • The function \( \tan^{-1}(x) \) will return an angle \( \theta \) such that \( \tan(\theta) = x \).
  • This function is restricted to angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) to ensure it gives a unique result.
When we are asked to evaluate \( \tan^{-1}(\sqrt{3}) \), we are essentially looking for an angle whose tangent is \( \sqrt{3} \). This is a crucial step in many trigonometry problems.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables within their domain. These identities are pivotal in solving trigonometric equations and simplifying expressions.The identity we focus on for this problem is the tangent of an angle at \( \frac{\pi}{3} \) (or 60 degrees), which equals \( \sqrt{3} \). Understanding this identity makes it easier to see that \( \tan^{-1}(\sqrt{3}) \) corresponds to \( \frac{\pi}{3} \). Here鈥檚 why:
  • Trigonometric identities help identify the tangent values of special angles (like \( \frac{\pi}{3} \) for \( \sqrt{3} \)).
  • Knowing these specific identities enables quick conversion between angles and their trigonometric ratios.
These relationships not only allow us to solve inverse trig functions but help in understanding the broader scope of trigonometry.
Radian Measure
Radian measure is another essential concept in understanding angles, frequently used in trigonometry, calculus, and physics because of its natural connection to arc length and sector area.But what exactly is a radian?
  • A radian is the angle created when the radius of a circle is wrapped along its circumference.
  • One full rotation in a circle is \( 2\pi \) radians, which corresponds to 360 degrees.
Understanding radians allows us to convert between degree measure and radian measure:
  • \( \pi \) radians equal 180 degrees.
  • Therefore, to convert radians to degrees, multiply by \( \frac{180}{\pi} \), and for degrees to radians, use \( \frac{\pi}{180} \).
In the context of this exercise, when we say \( \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \), \( \frac{\pi}{3} \) is an angle measured in radians, corresponding to 60 degrees. This conversion and understanding help solidify the use of radian measure as a default in many mathematical contexts.

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