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.

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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°.

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.

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.

All Flashcards

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°.

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.

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.