Textbook Question
In Exercises 7β12, find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s. Radius, r: 1 meter Arc Length, s: 600 centimeters
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 1, Problem 13
Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.
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tan π/3
Verified step by step guidance1
Recall that the angle \( \frac{\pi}{3} \) radians corresponds to 60 degrees, a common special angle in trigonometry.
Identify or recall the side lengths of a 30-60-90 right triangle, which are in the ratio 1 : \( \sqrt{3} \) : 2, where the side opposite 60 degrees (\( \frac{\pi}{3} \)) is \( \sqrt{3} \), the side opposite 30 degrees is 1, and the hypotenuse is 2.
Use the definition of tangent for an angle in a right triangle: \( \tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} \). For \( \theta = \frac{\pi}{3} \), the opposite side is \( \sqrt{3} \) and the adjacent side is 1.
Write the expression for \( \tan \frac{\pi}{3} \) as \( \frac{\sqrt{3}}{1} \).
Since the denominator is already rational (1), no rationalization is needed. The expression is simplified as is.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of the Tangent Function
The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the adjacent side. For angle ΞΈ, tan(ΞΈ) = opposite/adjacent. This ratio helps evaluate trigonometric expressions using triangle side lengths.
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Introduction to Tangent Graph
Special Angles and Their Trigonometric Values
Certain angles like Ο/3 (60Β°) have well-known exact trigonometric values. For Ο/3, tan(Ο/3) equals β3. Recognizing these special angles allows quick evaluation without needing a calculator or complex calculations.
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3:28Common Trig Functions For 45-45-90 Triangles
Rationalizing the Denominator
Rationalizing the denominator involves eliminating square roots from the denominator of a fraction by multiplying numerator and denominator by a suitable radical. This process simplifies expressions and is often required for final answers in trigonometry.
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2:58Rationalizing Denominators
Related Practice
Textbook Question
In Exercises 5β18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of
0, π, π, π, 2π, 5π, π, 7π, 4π, 3π, 5π, 11π, and 2π.
6 3 2 3 6 6 3 2 3 6
Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.
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In Exercises 11β18, continue to refer to the figure at the bottom of the previous page.
sec 11π/6
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Textbook Question
In Exercises 13β17, find a positive angle less than 360Β° or 2π that is coterminal with the given angle. -445Β°
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Textbook Question
In Exercises 5β18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of0, π, π, π, 2π, 5π, π, 7π, 4π, 3π, 5π, 11π, and 2π.6 3 2 3 6 6 3 2 3 6Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.In Exercises 11β18, continue to refer to the figure at the bottom of the previous page.sec 5π/3
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Textbook Question
In Exercises 5β18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0, π, π, π, 2π, 5π, π, 7π, 4π, 3π, 5π, 11π, and 2π. 6 3 2 3 6 6 3 2 3 6 Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.
In Exercises 11β18, continue to refer to the figure at the bottom of the previous page. csc 4π/3
Textbook Question
In Exercises 8β12, draw each angle in standard position. -135Β°
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