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Solve sin(x) = 0.8 for x in [0, 2\(\pi\)].
1. x = arcsin(0.8) ≈ 0.927 radians. 2. Other solution: \(\pi\) - 0.927 ≈ 2.214 radians.
Solve cos(x) = -0.5 for x in [0, 2\(\pi\)].
1. x = arccos(-0.5) = \(\frac{2\pi}{3}\). 2. Other solution: 2\(\pi\) - \(\frac{2\pi}{3}\) = \(\frac{4\pi}{3}\).
Solve tan(x) = 1 for x in [-\(\frac{\pi}{2}\), \(\frac{\pi}{2}\)].
1. x = arctan(1) = \(\frac{\pi}{4}\).
Solve sin(x) > 0.5 for x in [0, 2\(\pi\)].
1. Find where sin(x) = 0.5: x = \(\frac{\pi}{6}\), \(\frac{5\pi}{6}\). 2. sin(x) > 0.5 between these values: \(\frac{\pi}{6}\) < x < \(\frac{5\pi}{6}\).
Solve cos(x) < 0 for x in [0, 2\(\pi\)].
1. Find where cos(x) = 0: x = \(\frac{\pi}{2}\), \(\frac{3\pi}{2}\). 2. cos(x) < 0 between these values: \(\frac{\pi}{2}\) < x < \(\frac{3\pi}{2}\).
Solve tan(x) > 1 for x in [0, \(\pi\)].
1. Find where tan(x) = 1: x = \(\frac{\pi}{4}\). 2. tan(x) > 1 between \(\frac{\pi}{4}\) and \(\frac{\pi}{2}\): \(\frac{\pi}{4}\) < x < \(\frac{\pi}{2}\).
Solve 2sin(x) - 1 = 0 in the interval [0, 2\(\pi\)].
1. 2sin(x) = 1. 2. sin(x) = 1/2. 3. x = \(\frac{\pi}{6}\), \(\frac{5\pi}{6}\).
Solve 2sin(x) - 1 > 0 in the interval [0, 2\(\pi\)].
1. 2sin(x) > 1. 2. sin(x) > 1/2. 3. \(\frac{\pi}{6}\) < x < \(\frac{5\pi}{6}\).
Find all solutions to sin(x) = 0.6 in [0, 360°].
1. x = arcsin(0.6) ≈ 36.87°. 2. Other solution: 180° - 36.87° ≈ 143.13°.
Find all solutions to cos(x) > 0.5 in [0, 360°].
1. x = arccos(0.5) = 60°. 2. cos(x) > 0.5 when 0° ≤ x < 60° and 300° < x ≤ 360°.
Define arcsin(x).
The inverse sine function, giving the angle whose sine is x.
Define arccos(x).
The inverse cosine function, giving the angle whose cosine is x.
Define arctan(x).
The inverse tangent function, giving the angle whose tangent is x.
What is the domain of arcsin(x)?
[-1, 1]
What is the range of arcsin(x)?
[-\(\frac{\pi}{2}\), \(\frac{\pi}{2}\)]
What is the domain of arccos(x)?
[-1, 1]
What is the range of arccos(x)?
[0, \(\pi\)]
What is the domain of arctan(x)?
All real numbers
What is the range of arctan(x)?
(-\(\frac{\pi}{2}\), \(\frac{\pi}{2}\))
What does it mean for a trigonometric function to be periodic?
It repeats its values at regular intervals.
Explain the concept of inverse trigonometric functions.
Inverse trig functions 'undo' regular trig functions, helping find angles from ratios.
Explain the importance of domain restrictions when using inverse trig functions.
Domain restrictions ensure the inverse trig functions are well-defined and give unique outputs.
Why do trigonometric equations often have multiple solutions?
Trigonometric functions are periodic, repeating their values at regular intervals.
How does the unit circle help in solving trigonometric equations and inequalities?
It provides a visual representation of trig function values for different angles.
What is the relationship between sine and arcsine?
arcsin(x) gives the angle whose sine is x. They are inverse functions.
What is the relationship between cosine and arccosine?
arccos(x) gives the angle whose cosine is x. They are inverse functions.
What is the relationship between tangent and arctangent?
arctan(x) gives the angle whose tangent is x. They are inverse functions.
Explain how to find all solutions to sin(x) = a.
Find the principal value using arcsin, then use periodicity to find other solutions.
Explain how to find all solutions to cos(x) = a.
Find the principal value using arccos, then use periodicity to find other solutions.
Explain how to find all solutions to tan(x) = a.
Find the principal value using arctan, then use periodicity to find other solutions.