Unlock the Unit Circle: Mastering Inverse Trigonometric Functions

Algebra 2 Grades High School 8:56 Video
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Lesson Description

Explore inverse trigonometric functions (arcsin, arccos, arctan) using the unit circle. Learn about restricted domains and practice with examples.

Video Resource

Evaluating Inverse Trigonometric Functions (arcsin, arccos, arctan) Using Unit Circle

Mario's Math Tutoring

Duration: 8:56
Watch on YouTube

Key Concepts

  • Inverse trigonometric functions (arcsin, arccos, arctan)
  • Unit circle
  • Restricted domains of inverse trigonometric functions

Learning Objectives

  • Evaluate inverse trigonometric functions using the unit circle.
  • Identify the restricted domains of arcsin, arccos, and arctan.
  • Apply the restricted domains to find the correct angle when evaluating inverse trigonometric functions.

Educator Instructions

  • Introduction (5 mins)
    Briefly review trigonometric functions and the unit circle. Introduce the concept of inverse trigonometric functions as finding the angle given a trigonometric value. Emphasize that we are finding angles (in radians).
  • Understanding Restricted Domains (10 mins)
    Explain why inverse trigonometric functions have restricted domains. Use the sine, cosine, and tangent graphs to illustrate the need for restrictions to ensure the inverse is a function (passes the horizontal line test). Specify the restricted domains for arcsin [-π/2, π/2], arccos [0, π], and arctan (-π/2, π/2).
  • Evaluating Inverse Trig Functions Using the Unit Circle (20 mins)
    Guide students through examples of evaluating arcsin, arccos, and arctan using the unit circle. Emphasize identifying the correct quadrant based on the restricted domain and the sign of the trigonometric value. Work through examples from the video, pausing to ask students for input.
  • Practice Problems (15 mins)
    Have students work individually or in pairs on practice problems. Circulate to provide assistance. Use examples not already covered in the video. Encourage the use of the unit circle as a visual aid.
  • Wrap-up and Review (10 mins)
    Review key concepts, focusing on the importance of restricted domains. Address any remaining student questions. Preview future topics related to trigonometry.

Interactive Exercises

  • Unit Circle Scavenger Hunt
    Give students a list of inverse trigonometric function values (e.g., arcsin(1/2), arccos(-1), arctan(1)). Have them find the corresponding angles on a physical or digital unit circle.

Discussion Questions

  • Why do we need to restrict the domains of inverse trigonometric functions?
  • How does the unit circle help us evaluate inverse trigonometric functions?
  • What are the restricted domains for arcsin, arccos, and arctan?

Skills Developed

  • Evaluating inverse trigonometric functions
  • Applying knowledge of the unit circle
  • Understanding restricted domains
  • Problem-solving

Multiple Choice Questions

Question 1:

What is the restricted domain for arcsin(x)?

Correct Answer: [-π/2, π/2]

Question 2:

What is the value of arccos(-1)?

Correct Answer: π

Question 3:

What quadrant(s) are used when finding arctan?

Correct Answer: I and IV

Question 4:

What is arcsin(√3/2)?

Correct Answer: π/3

Question 5:

Which trigonometric function corresponds to the x-coordinate on the unit circle?

Correct Answer: Cosine

Question 6:

arctan(y/x) is another way of saying:

Correct Answer: arctangent

Question 7:

Where is cosine equal to zero on the unit circle?

Correct Answer: π/2

Question 8:

What is the restricted domain for arccos(x)?

Correct Answer: [0, π]

Question 9:

arctan(1) is equal to what?

Correct Answer: π/4

Question 10:

The unit circle is used to help evaluate inverse trig functions because:

Correct Answer: It has all the trig values

Fill in the Blank Questions

Question 1:

The restricted domain of arcsin(x) is [____, ____].

Correct Answer: -π/2, π/2

Question 2:

The value of arccos(0) is ____.

Correct Answer: π/2

Question 3:

arctan(____) = 0

Correct Answer: 0

Question 4:

arcsin(1) = ____.

Correct Answer: π/2

Question 5:

arccos(1/2) = ____.

Correct Answer: π/3

Question 6:

arctan(√3) = ____.

Correct Answer: π/3

Question 7:

The range of arccos is ____.

Correct Answer: [0, π]

Question 8:

arcsin(____) = -π/6

Correct Answer: -1/2

Question 9:

arctan is y/x, therefore it is also _____/_____

Correct Answer: sine, cosine

Question 10:

The restricted domain for tangent is (_____, _____)

Correct Answer: -π/2, π/2

Educational Standards

A2.F.1.1:Use algebraic, interval, and set notations to specify the domain and range of various types of functions, and evaluate a function at a given point in its domain. A2.F.1:Understand functions as descriptions of covariation (how related quantities vary together).

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