Graphing Arcsine: Unveiling the Inverse of Sine

PreAlgebra Grades High School 3:22 Video
Print Lesson Plan Watch Video

Lesson Description

Explore the concept of inverse trigonometric functions by focusing on graphing arcsine (the inverse of sine). Learn about domain restriction, reflection over y=x, and key points on the graph.

Video Resource

How to Graph Arcsin (sine inverse)

Mario's Math Tutoring

Duration: 3:22
Watch on YouTube

Key Concepts

  • Inverse Functions
  • Domain Restriction
  • Reflection over y=x
  • Arcsine (Sine Inverse)

Learning Objectives

  • Explain why the sine function needs a restricted domain to have an inverse.
  • Graph the arcsine function using the reflection method over the line y=x.
  • Identify key points on the arcsine graph and relate them to the sine graph.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of inverse functions and the horizontal line test. Briefly discuss why not all functions have an inverse. Introduce the concept of trigonometric functions and lead into the video.
  • Video Viewing (7 mins)
    Play the YouTube video 'How to Graph Arcsin (sine inverse)' by Mario's Math Tutoring. Encourage students to take notes on key concepts like domain restriction and the reflection method.
  • Guided Practice (15 mins)
    Work through examples similar to those in the video. Focus on identifying key points on the sine graph, switching the x and y coordinates, and plotting the corresponding points on the arcsine graph. Emphasize the reflection over y=x.
  • Independent Practice (10 mins)
    Have students work individually on graphing additional arcsine functions or variations thereof (e.g., arcsin(x) + 1, 2*arcsin(x)). Circulate to provide assistance as needed.
  • Wrap-up and Discussion (3 mins)
    Summarize the key concepts and address any remaining questions. Preview the next lesson on other inverse trigonometric functions.

Interactive Exercises

  • Graphing Arcsine with GeoGebra
    Use GeoGebra to graph both the sine function (with restricted domain) and the arcsine function. Experiment with reflecting points and the entire graph over the line y=x.
  • Coordinate Matching
    Provide students with a set of coordinate pairs from the restricted sine function and ask them to match each pair with its corresponding coordinate pair on the arcsine function.

Discussion Questions

  • Why is it necessary to restrict the domain of the sine function before finding its inverse?
  • How does reflecting the graph of sine over the line y=x help us visualize the graph of arcsine?
  • What are the key differences between the graph of sine and the graph of arcsine?

Skills Developed

  • Graphing Functions
  • Understanding Inverse Functions
  • Applying Domain Restrictions
  • Visualizing Transformations

Multiple Choice Questions

Question 1:

Why is the domain of the sine function restricted when finding the arcsine function?

Correct Answer: To make the sine function pass the horizontal line test.

Question 2:

Over what line is the graph of the sine function (with restricted domain) reflected to obtain the graph of the arcsine function?

Correct Answer: y = x

Question 3:

What is the domain of the arcsine function?

Correct Answer: [-1, 1]

Question 4:

What is the range of the arcsine function?

Correct Answer: [-π/2, π/2]

Question 5:

If the point (π/2, 1) lies on the graph of the sine function, what point lies on the graph of the arcsine function?

Correct Answer: (1, π/2)

Question 6:

Which of the following is equivalent to arcsin(x)?

Correct Answer: sin⁻¹(x)

Question 7:

The restricted domain of sin(x) used to define arcsin(x) is:

Correct Answer: [-π/2, π/2]

Question 8:

The arcsine function is the inverse of the sine function only on a specific interval. This interval is crucial for:

Correct Answer: Simplifying calculations.

Question 9:

Which of the following statements is true about the relationship between sin(x) and arcsin(x)?

Correct Answer: They are reflections across the line y=x.

Question 10:

What is the value of arcsin(0)?

Correct Answer: 0

Fill in the Blank Questions

Question 1:

The process of switching the x and y coordinates when graphing an inverse function is equivalent to reflecting the graph over the line y = ______.

Correct Answer: x

Question 2:

The arcsine function is also known as the _______ of sine.

Correct Answer: inverse

Question 3:

To make the sine function invertible, we _________ its domain.

Correct Answer: restrict

Question 4:

The domain of the arcsine function is [______, ______].

Correct Answer: -1, 1

Question 5:

The range of the arcsine function is [______, ______].

Correct Answer: -π/2, π/2

Question 6:

The value of arcsin(1) is _______.

Correct Answer: π/2

Question 7:

The restricted domain of sin(x) for arcsin(x) is from -π/2 to ________.

Correct Answer: π/2

Question 8:

The graph of y=arcsin(x) is a __________ of y=sin(x) across the line y=x.

Correct Answer: reflection

Question 9:

If sin(π/4) = √2/2, then arcsin(√2/2) = _______.

Correct Answer: π/4

Question 10:

The arcsin function only outputs angles between -π/2 and π/2, these angles are found in quadrants ______ and ______.

Correct Answer: IV, I

Educational Standards

PC.F.1.5:Determine if a function has an inverse. Algebraically and graphically find the inverse or define any restrictions on the domain that meet the requirement for invertibility, and find the inverse on the restricted domain. PC.T.1.8:Graph all six trigonometric functions, identifying key features. PC.F.1.2:Sketch the graph of a function that models a relationship between two quantities, identifying key features.

Teaching Materials

Download ready-to-use materials for this lesson:

Student Worksheet Classroom Slides Assessment Quiz
Add to Google Classroom

User Actions

Sign in to save this lesson plan to your favorites.

Sign In

Share This Lesson

Related Lesson Plans