Textbook Question
In Exercises 35β60, find the reference angle for each angle. -13π/3
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 1, Problem 1.2.63
In Exercises 63β68, find the exact value of each expression. Do not use a calculator. tan(π/3)/2 - 1/sec(π/6)
Verified step by step guidance1
Identify the given expression: \(\frac{\tan \frac{\pi}{3} - 1}{2 \sec \frac{\pi}{6}}\).
Recall the exact values of the trigonometric functions involved: \(\tan \frac{\pi}{3} = \sqrt{3}\), and \(\sec \frac{\pi}{6} = \frac{1}{\cos \frac{\pi}{6}}\).
Find \(\cos \frac{\pi}{6}\) using the known exact value: \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\), so \(\sec \frac{\pi}{6} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}\).
Substitute these values back into the expression: \(\frac{\sqrt{3} - 1}{2 \times \frac{2}{\sqrt{3}}}\).
Simplify the denominator and then the entire fraction step-by-step, rationalizing denominators if necessary, to find the exact value.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exact Values of Special Angles
Certain angles like Ο/3 and Ο/6 have well-known exact trigonometric values derived from special triangles. For example, tan(Ο/3) equals β3, and sec(Ο/6) equals 2/β3. Knowing these values allows precise calculation without a calculator.
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45-45-90 Triangles
Definition and Relationship of Trigonometric Functions
Understanding how tangent and secant relate to sine and cosine is essential. Tangent is the ratio of sine to cosine (tan ΞΈ = sin ΞΈ / cos ΞΈ), while secant is the reciprocal of cosine (sec ΞΈ = 1 / cos ΞΈ). This helps in simplifying and evaluating expressions.
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Simplifying Trigonometric Expressions
Combining and simplifying expressions involving trigonometric functions requires algebraic manipulation, such as finding common denominators or rationalizing denominators. This skill ensures the exact value is expressed in simplest form without approximations.
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6:36Simplifying Trig Expressions
Related Practice
Textbook Question
In Exercises 57β70, find a positive angle less than or that is coterminal with the given angle. -760Β°
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Textbook Question
In Exercises 63β68, find the exact value of each expression. Do not use a calculator. 1 + sinΒ² 40Β° + sinΒ² 50Β°
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Textbook Question
In Exercises 35β60, find the reference angle for each angle. 17π / 6
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Textbook Question
In Exercises 41β56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.
120Β°
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Textbook Question
In Exercises 57β70, find a positive angle less than or that is coterminal with the given angle. 395Β°
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