Textbook Question
Determine whether each statement is true or false. If false, tell why. See Example 4.tan² 60° + 1 = sec² 60°
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3. Unit Circle
Reference Angles
3:47 minutesProblem 68
Textbook Question
Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. See Example 6. √3sin θ = - —— 2
Verified step by step guidance1
Recognize that the equation \( \sin \theta = -\frac{\sqrt{3}}{2} \) corresponds to a known sine value. The positive value \( \sin \theta = \frac{\sqrt{3}}{2} \) is associated with angles \( 60^\circ \) and \( 120^\circ \) in the unit circle.
Since the sine function is negative in the third and fourth quadrants, we need to find the reference angles in these quadrants.
In the third quadrant, the angle is \( 180^\circ + 60^\circ = 240^\circ \).
In the fourth quadrant, the angle is \( 360^\circ - 60^\circ = 300^\circ \).
Thus, the values of \( \theta \) that satisfy \( \sin \theta = -\frac{\sqrt{3}}{2} \) in the interval \([0^\circ, 360^\circ)\) are \( 240^\circ \) and \( 300^\circ \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Understanding the Unit Circle
The unit circle is a fundamental concept in trigonometry that defines the relationship between angles and the coordinates of points on the circle. Angles are measured from the positive x-axis, and the sine and cosine of an angle correspond to the y and x coordinates of a point on the circle, respectively. This understanding is crucial for determining the values of θ that satisfy trigonometric equations.
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Sine Function and Its Values
The sine function, denoted as sin(θ), represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. It is periodic and takes values between -1 and 1. In this problem, we need to find angles θ where sin(θ) equals a specific negative value, which indicates that θ must be in the third or fourth quadrants of the unit circle.
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Solving Trigonometric Equations
Solving trigonometric equations involves finding all angles that satisfy a given equation within a specified interval. This often requires using inverse trigonometric functions and understanding the periodic nature of trigonometric functions. In this case, we will find the reference angle for sin(θ) = -√3/2 and then determine the corresponding angles in the appropriate quadrants.
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