Textbook Question
Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. 8440°
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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 95
Concept Check Suppose that 90° < θ < 180° . Find the sign of each function value. cos ( ―θ)
Verified step by step guidance1
Recall the given range for \( \theta \): \( 90^\circ < \theta < 180^\circ \). This means \( \theta \) is in the second quadrant.
Understand that \( -\theta \) is the negative of \( \theta \). Since \( \theta \) is between 90° and 180°, \( -\theta \) will be between \( -180^\circ \) and \( -90^\circ \), which places \( -\theta \) in the third or fourth quadrant depending on the exact value.
Use the even-odd properties of cosine: cosine is an even function, so \( \cos(-\theta) = \cos(\theta) \). This means the sign of \( \cos(-\theta) \) is the same as the sign of \( \cos(\theta) \).
Determine the sign of \( \cos(\theta) \) in the second quadrant. Since cosine corresponds to the x-coordinate on the unit circle, and in the second quadrant x-values are negative, \( \cos(\theta) < 0 \).
Conclude that \( \cos(-\theta) \) is also negative because it equals \( \cos(\theta) \), which is negative in the second quadrant.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quadrants and Angle Ranges
The coordinate plane is divided into four quadrants, each with specific angle ranges and sign conventions for trigonometric functions. For angles between 90° and 180°, the angle lies in the second quadrant, where sine is positive and cosine is negative. Understanding the quadrant helps determine the sign of trigonometric values.
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Quadratic Formula
Even-Odd Identities in Trigonometry
Trigonometric functions have parity properties: cosine is an even function, meaning cos(-θ) = cos(θ), while sine is odd, so sin(-θ) = -sin(θ). This identity allows simplification of expressions involving negative angles by relating them to positive angles.
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Sign of Cosine Function in Different Quadrants
The cosine function's sign depends on the quadrant of the angle. It is positive in the first and fourth quadrants and negative in the second and third quadrants. Since θ is between 90° and 180°, cos(θ) is negative, and by the even property, cos(-θ) shares the same value and sign.
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06:14Sum and Difference of Sine & Cosine
Related Practice
Textbook Question
Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. -5280°
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Textbook Question
Give two positive and two negative angles that are coterminal with the given quadrantal angle. 90°
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Textbook Question
Use trigonometric function values of quadrantal angles to evaluate each expression. [cos(―180°)]² + [sin(― 180°)]²
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Textbook Question
Use trigonometric function values of quadrantal angles to evaluate each expression. ―3(sin 90°)⁴ + 4(cos 180°)³
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Textbook Question
Use trigonometric function values of quadrantal angles to evaluate each expression. (sec 180°)² ― 3 (sin 360°)² + cos 180°
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