Textbook Question
Find the values of the six trigonometric functions for an angle in standard position having each given point on its terminal side. Rationalize denominators when applicable. (―8 , 15)
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Ch. 1 - Trigonometric Functions
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 2, Problem 28
Determine the signs of the trigonometric functions of an angle in standard position with the given measure. See Example 2. 84°
Verified step by step guidance1
Identify the quadrant in which the angle 84° lies. Since 84° is between 0° and 90°, it is in the first quadrant.
Recall the sign rules for trigonometric functions in each quadrant: In the first quadrant, all trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) are positive.
Therefore, for the angle 84°, sine (sin 84°) is positive because sine corresponds to the y-coordinate on the unit circle, which is positive in the first quadrant.
Cosine (cos 84°) is positive because cosine corresponds to the x-coordinate on the unit circle, which is positive in the first quadrant.
Tangent (tan 84°), being the ratio of sine to cosine, is also positive in the first quadrant since both sine and cosine are positive.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Position of an Angle
An angle in standard position has its vertex at the origin and its initial side along the positive x-axis. The terminal side rotates counterclockwise for positive angles. Understanding this helps locate the angle's terminal side in the coordinate plane, which is essential for determining the signs of trigonometric functions.
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Quadrants and Sign of Trigonometric Functions
The coordinate plane is divided into four quadrants, each with specific signs for sine, cosine, and tangent. For example, in the first quadrant (0° to 90°), all trigonometric functions are positive. Knowing the quadrant of the angle helps identify the sign of each function.
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Evaluating Trigonometric Functions at Specific Angles
To determine the sign of trigonometric functions for a given angle, first find the quadrant of the angle, then apply the sign rules for sine, cosine, and tangent in that quadrant. For 84°, which lies in the first quadrant, sine, cosine, and tangent are all positive.
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Related Practice
Textbook Question
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29.6° , 49.7°
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Find the measure of each marked angle. See Example 2.
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Textbook Question
Find the measure of each marked angle. See Example 2.
1
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Textbook Question
The measures of two angles of a triangle are given. Find the measure of the third angle. See Example 2.
17° 41' 13" , 96° 12' 10"