Textbook Question
Concept Check Classify each triangle as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.
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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 34
Determine the signs of the trigonometric functions of an angle in standard position with the given measure. See Example 2.
―15°
Verified step by step guidance1
Identify the quadrant in which the angle lies. Since the angle is -15°, it is a negative angle measured clockwise from the positive x-axis. To find its equivalent positive angle, add 360°: \(-15° + 360° = 345°\). This places the angle in the fourth quadrant.
Recall the signs of the trigonometric functions in each quadrant. In the fourth quadrant: sine is negative, cosine is positive, and tangent is negative.
Determine the reference angle. The reference angle is the acute angle the terminal side makes with the x-axis. For 345°, the reference angle is \(360° - 345° = 15°\).
Use the reference angle to find the values of the trigonometric functions, keeping in mind their signs in the fourth quadrant. For example, \(\sin(-15°) = -\sin(15°)\), \(\cos(-15°) = \cos(15°)\), and \(\tan(-15°) = -\tan(15°)\).
Summarize the signs: sine is negative, cosine is positive, tangent is negative for the angle -15°.
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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 from the initial side by the given angle measure, positive for counterclockwise and negative for clockwise rotation. Understanding this helps locate the angle on the coordinate plane.
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Quadrants and Sign of Coordinates
The coordinate plane is divided into four quadrants, each determining the signs of x and y coordinates. Since trigonometric functions depend on these coordinates, knowing the quadrant of the terminal side is essential to determine the sign of sine, cosine, and tangent.
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Signs of Trigonometric Functions by Quadrant
Sine, cosine, and tangent functions have specific signs in each quadrant: sine is positive in QI and QII, cosine in QI and QIV, and tangent in QI and QIII. Identifying the quadrant of the angle allows you to assign the correct sign to each trigonometric function.
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Related Practice
Textbook Question
Find the measure of the smaller angle formed by the hands of a clock at the following times. 3:15
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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.) IV , x/y
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Textbook Question
Find the measure of the smaller angle formed by the hands of a clock at the following times.
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Textbook Question
An equation of the terminal side of an angle θ in standard position is given with a restriction on x. Sketch the least positive such angle θ , and find the values of the six trigonometric functions of θ . 12x + 5y = 0 , x ≥ 0 .
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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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