Textbook Question
Determine whether each statement is true or false. If false, tell why. See Example 4.cos(30° + 60°) = cos 30° + cos 60°
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3. Unit Circle
Reference Angles
3:28 minutesProblem 64
Textbook Question
Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. See Example 6.sec θ = -√2
Verified step by step guidance1
Recognize that \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, \( \sec \theta = -\sqrt{2} \) implies \( \cos \theta = -\frac{1}{\sqrt{2}} \).
Simplify \( \cos \theta = -\frac{1}{\sqrt{2}} \) to \( \cos \theta = -\frac{\sqrt{2}}{2} \).
Recall that \( \cos \theta = -\frac{\sqrt{2}}{2} \) corresponds to angles where the cosine value is negative, which occurs in the second and third quadrants.
Identify the reference angle where \( \cos \theta = \frac{\sqrt{2}}{2} \), which is \( 45^\circ \).
Determine the angles in the second and third quadrants: \( \theta = 180^\circ - 45^\circ \) and \( \theta = 180^\circ + 45^\circ \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Secant Function
The secant function, denoted as sec(θ), is the reciprocal of the cosine function. It is defined as sec(θ) = 1/cos(θ). Understanding the secant function is crucial for solving equations involving it, as it helps identify the relationship between secant and cosine values, particularly in determining the angles that yield specific secant values.
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Quadrants and Sign of Trigonometric Functions
Trigonometric functions have different signs in different quadrants of the unit circle. In the case of sec(θ) = -√2, we need to identify in which quadrants the secant function is negative. Since sec(θ is negative in the second and third quadrants, this knowledge is essential for finding the correct angle solutions within the specified interval.
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Finding Angles from Trigonometric Values
To find angles corresponding to a specific trigonometric value, one must use the inverse functions and consider the periodic nature of trigonometric functions. For sec(θ) = -√2, we first find the reference angle where sec(θ) = √2, then apply the appropriate quadrant adjustments to find all angles θ in the interval [0°, 360°) that satisfy the equation.
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