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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°.
How does the graph of y = sin(x) relate to the values of arcsin(x)?
arcsin(x) gives the x-value (angle) on the sine graph for a given y-value.
How does the graph of y = cos(x) relate to the values of arccos(x)?
arccos(x) gives the x-value (angle) on the cosine graph for a given y-value.
How does the graph of y = tan(x) relate to the values of arctan(x)?
arctan(x) gives the x-value (angle) on the tangent graph for a given y-value.
What does the graph of y = arcsin(x) look like?
It's the reflection of y = sin(x) across y = x, restricted to [-1, 1] on the x-axis.
What does the graph of y = arccos(x) look like?
It's the reflection of y = cos(x) across y = x, restricted to [-1, 1] on the x-axis.
What does the graph of y = arctan(x) look like?
It's the reflection of y = tan(x) across y = x, with horizontal asymptotes at y = ±\(\frac{\pi}{2}\).
How can you use the graph of sin(x) to solve sin(x) > a?
Find where sin(x) = a, then identify the intervals where the graph is above y = a.
How can you use the graph of cos(x) to solve cos(x) < a?
Find where cos(x) = a, then identify the intervals where the graph is below y = a.
How can you use the graph of tan(x) to solve tan(x) > a?
Find where tan(x) = a, then identify the intervals where the graph is above y = a.
What does the x-intercept of f(x) = 2sin(x) - 1 represent?
It represents the solutions to the equation 2sin(x) - 1 = 0.
General solution for sin(x) = a.
x = arcsin(a) + 2\(\pi\)k or x = \(\pi\) - arcsin(a) + 2\(\pi\)k, where k is an integer.
General solution for cos(x) = a.
x = arccos(a) + 2\(\pi\)k or x = -arccos(a) + 2\(\pi\)k, where k is an integer.
General solution for tan(x) = a.
x = arctan(a) + \(\pi\)k, where k is an integer.
How do you find the solutions to sin(x) = a in the interval [0, 2\(\pi\)]?
Find arcsin(a). If arcsin(a) is in the interval, it's one solution. The other is \(\pi\) - arcsin(a).
How do you find the solutions to cos(x) = a in the interval [0, 2\(\pi\)]?
Find arccos(a). If arccos(a) is in the interval, it's one solution. The other is 2\(\pi\) - arccos(a).
How do you find the solutions to tan(x) = a in the interval [0, \(\pi\)]?
Find arctan(a). If arctan(a) is in the interval, it's the solution.
What is the range of the arcsine function?
[-\(\frac{\pi}{2}\), \(\frac{\pi}{2}\)]
What is the range of the arccosine function?
[0, \(\pi\)]
What is the range of the arctangent function?
(-\(\frac{\pi}{2}\), \(\frac{\pi}{2}\))
If sin(x) = 0.5, what is x?
x = arcsin(0.5) = \(\frac{\pi}{6}\) or 30°