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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 = ± $\frac{\pi}{2}$ .

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°

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

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 $\pi$ ].

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 [- $\frac{\pi}{2}$ , $\frac{\pi}{2}$ ].

x = arctan(1) = $\frac{\pi}{4}$ .

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

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 $\pi$ ].

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, $\pi$ ].

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 $\pi$ ].

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 $\pi$ ].

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°].

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°].

x = arccos(0.5) = 60°. 2. cos(x) > 0.5 when 0° ≤ x < 60° and 300° < x ≤ 360°.

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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 = ± $\frac{\pi}{2}$ .

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°

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

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 $\pi$ ].

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 [- $\frac{\pi}{2}$ , $\frac{\pi}{2}$ ].

x = arctan(1) = $\frac{\pi}{4}$ .

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

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 $\pi$ ].

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, $\pi$ ].

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 $\pi$ ].

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 $\pi$ ].

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°].

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°].

x = arccos(0.5) = 60°. 2. cos(x) > 0.5 when 0° ≤ x < 60° and 300° < x ≤ 360°.