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  1. AP Pre Calculus
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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.

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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 = ±π2\frac{\pi}{2}2π​.

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) + 2π\piπk or x = π\piπ - arcsin(a) + 2π\piπk, where k is an integer.

General solution for cos(x) = a.

x = arccos(a) + 2π\piπk or x = -arccos(a) + 2π\piπk, where k is an integer.

General solution for tan(x) = a.

x = arctan(a) + π\piπk, where k is an integer.

How do you find the solutions to sin(x) = a in the interval [0, 2π\piπ]?

Find arcsin(a). If arcsin(a) is in the interval, it's one solution. The other is π\piπ - arcsin(a).

How do you find the solutions to cos(x) = a in the interval [0, 2π\piπ]?

Find arccos(a). If arccos(a) is in the interval, it's one solution. The other is 2π\piπ - arccos(a).

How do you find the solutions to tan(x) = a in the interval [0, π\piπ]?

Find arctan(a). If arctan(a) is in the interval, it's the solution.

What is the range of the arcsine function?

[-π2\frac{\pi}{2}2π​, π2\frac{\pi}{2}2π​]

What is the range of the arccosine function?

[0, π\piπ]

What is the range of the arctangent function?

(-π2\frac{\pi}{2}2π​, π2\frac{\pi}{2}2π​)

If sin(x) = 0.5, what is x?

x = arcsin(0.5) = π6\frac{\pi}{6}6π​ or 30°

Solve sin(x) = 0.8 for x in [0, 2π\piπ].

  1. x = arcsin(0.8) ≈ 0.927 radians. 2. Other solution: π\piπ - 0.927 ≈ 2.214 radians.

Solve cos(x) = -0.5 for x in [0, 2π\piπ].

  1. x = arccos(-0.5) = 2π3\frac{2\pi}{3}32π​. 2. Other solution: 2π\piπ - 2π3\frac{2\pi}{3}32π​ = 4π3\frac{4\pi}{3}34π​.

Solve tan(x) = 1 for x in [-π2\frac{\pi}{2}2π​, π2\frac{\pi}{2}2π​].

  1. x = arctan(1) = π4\frac{\pi}{4}4π​.

Solve sin(x) > 0.5 for x in [0, 2π\piπ].

  1. Find where sin(x) = 0.5: x = π6\frac{\pi}{6}6π​, 5π6\frac{5\pi}{6}65π​. 2. sin(x) > 0.5 between these values: π6\frac{\pi}{6}6π​ < x < 5π6\frac{5\pi}{6}65π​.

Solve cos(x) < 0 for x in [0, 2π\piπ].

  1. Find where cos(x) = 0: x = π2\frac{\pi}{2}2π​, 3π2\frac{3\pi}{2}23π​. 2. cos(x) < 0 between these values: π2\frac{\pi}{2}2π​ < x < 3π2\frac{3\pi}{2}23π​.

Solve tan(x) > 1 for x in [0, π\piπ].

  1. Find where tan(x) = 1: x = π4\frac{\pi}{4}4π​. 2. tan(x) > 1 between π4\frac{\pi}{4}4π​ and π2\frac{\pi}{2}2π​: π4\frac{\pi}{4}4π​ < x < π2\frac{\pi}{2}2π​.

Solve 2sin(x) - 1 = 0 in the interval [0, 2π\piπ].

  1. 2sin(x) = 1. 2. sin(x) = 1/2. 3. x = π6\frac{\pi}{6}6π​, 5π6\frac{5\pi}{6}65π​.

Solve 2sin(x) - 1 > 0 in the interval [0, 2π\piπ].

  1. 2sin(x) > 1. 2. sin(x) > 1/2. 3. π6\frac{\pi}{6}6π​ < x < 5π6\frac{5\pi}{6}65π​.

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°.