Textbook Question
Find the measure of each marked angle. See Example 2 complementary angles with measures 3𝓍 ― 5 and 6𝓍 ― 40 degrees
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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 31
Determine the signs of the trigonometric functions of an angle in standard position with the given measure. See Example 2.
178°
Verified step by step guidance1
Identify the quadrant in which the angle 178° lies. Since 178° is between 90° and 180°, it lies in the second quadrant.
Recall the signs of the trigonometric functions in the second quadrant: sine (sin) is positive, cosine (cos) is negative, and tangent (tan) is negative.
Use the reference angle to understand the values better. The reference angle for 178° is calculated as \(180^\circ - 178^\circ = 2^\circ\).
Determine the signs of the reciprocal functions based on the signs of sine, cosine, and tangent: cosecant (csc) has the same sign as sine, secant (sec) has the same sign as cosine, and cotangent (cot) has the same sign as tangent.
Summarize the signs: sin(178°) > 0, cos(178°) < 0, tan(178°) < 0, csc(178°) > 0, sec(178°) < 0, cot(178°) < 0.
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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 on the coordinate plane, which is essential for determining the signs of trigonometric functions.
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Drawing Angles in Standard Position
Quadrants and Sign of Trigonometric Functions
The coordinate plane is divided into four quadrants, each determining the sign of sine, cosine, and tangent. For example, in the second quadrant (90° to 180°), sine is positive while cosine and tangent are negative. Knowing the quadrant of the angle helps identify the signs of its trigonometric values.
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6:36Quadratic Formula
Reference Angle
The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. It helps find the exact values of trigonometric functions by relating them to known angles in the first quadrant, while the quadrant determines the sign. For 178°, the reference angle is 2°.
Recommended video:
5:31Reference Angles on the Unit Circle
Related Practice
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. (6√3 , ―6)
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Textbook Question
Find the measure of each marked angle. See Example 2 complementary angles with measures 9𝓍 + 6 and 3𝓍 degrees
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Textbook Question
Find the measure of each marked angle. See Example 2 supplementary angles with measures 6𝓍 - 4 and 8𝓍 - 12 degrees
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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"
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Textbook Question
Concept Check Suppose that the point (x, y) is in the indicated quadrant. Determine whether the given ratio is positive or negative. Recall that r = √(x² + y²) .(Hint: Drawing a sketch may help.) III , y/r
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