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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.

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.

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°.