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Chapter 1: Problem 7

\(\tan 30^{\circ}\)

Short Answer

Expert verified
\(\tan 30^{\circ} = \frac{\sqrt{3}}{3}\)

Step by step solution

01

Identify the Problem

We need to find the value of the tangent of the angle 30 degrees, which is written as \(\tan 30^{\circ}\).
02

Recall Trigonometric Values

Recall the trigonometric values for standard angles. For \(30^{\circ}\), the value for tangent is \(\tan 30^{\circ} = \frac{1}{\sqrt{3}}\).
03

Simplify the Expression

If needed, simplify the expression \(\frac{1}{\sqrt{3}}\) to a more standard form by rationalizing the denominator. Multiply the numerator and the denominator by \(\sqrt{3}\) to obtain \(\frac{\sqrt{3}}{3}\).

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Key Concepts

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

Tangent Function
Tangent is one of the primary trigonometric functions, alongside sine and cosine. It is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. This is expressed as:
  • tan(胃) = \(\frac{\text{opposite}}{\text{adjacent}}\)
Alternatively, in terms of sine and cosine, the tangent function can be expressed as:
  • tan(胃) = \(\frac{\sin(胃)}{\cos(胃)}\)
This dual definition helps in solving a variety of problems due to its flexibility. When dealing with the tangent function, it is important to remember that it has properties such as periodicity and asymptotes. Asymptotes occur where the cosine of the angle is zero since division by zero is undefined.
Standard Angles
Standard angles are specific angles that are frequently used in trigonometry, often simplifying calculations. Angles such as 0掳, 30掳, 45掳, 60掳, and 90掳 are considered standard. These angles have known, exact trigonometric values, making them useful study aids and solutions to many problems.
  • For example, the trigonometric values for 30掳 include:
    • \(\sin(30^{\circ}) = \frac{1}{2}\)
    • \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\)
    • \(\tan(30^{\circ}) = \frac{1}{\sqrt{3}}\)
Knowing these values allows for quick calculations without extensive computation. Moreover, understanding these values for standard angles provides a solid foundation for predicting the behavior of trigonometric functions at non-standard angles.
Rationalizing Denominators
Rationalizing the denominator is a technique used to eliminate square roots or other irrational numbers from the denominator of a fraction. This is achieved by multiplying both the top and bottom of the fraction by a suitable expression. For instance, given a fraction \(\frac{1}{\sqrt{3}}\), to rationalize the denominator, you multiply the numerator and the denominator by \(\sqrt{3}\):
  • \(\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}\)
Rationalizing helps in simplifying expressions and is often required in mathematical fields such as calculus and algebra for clean, manageable results. It also ensures that expressions conform to a standard mathematical format, making them easier to read and interpret.

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Most popular questions from this chapter

In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, find the length of the side opposite the \(60^{\circ}\) angle if the side opposite the \(30^{\circ}\) angle is 10 inches. Solution: The length opposite the \(60^{\circ}\) angle is twice the length opposite the \(30^{\circ}\) angle. \(2(10)=20\) The side opposite the \(60^{\circ}\) angle has length 20 inches. This is incorrect. What mistake was made?

\(\tan 54^{\circ}\)

Golf. If the flagpole that a golfer aims at on a green measures 5 feet from the ground to the top of the flag and a golfer measures a \(3^{\circ}\) angle from top to bottom of the pole, how far (in horizontal distance) is the golfer from the flag? Round to the nearest foot.

\(\sec 75^{\circ}\)

\(\csc 51^{\circ}\)
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