Inverse Trig Ratios (Lesson 4.2)
Unit 0: Prerequisites
Day 1: The Cartesian Plane
Day 2: Equations of Circles
Day 3: Solving Equations in Multiple Representations
Day 4: Reasoning with Formulas
Day 5: Quiz 0.1 to 0.4
Day 6: Linear Relationships
Day 7: Reasoning with Slope
Day 8: Set Notation
Day 9: Quiz 0.5 to 0.7
Day 10: Unit 0 Review
Day 11: Unit 0 Test
Unit 1: Functions
Day 1: Functions and Function Notation
Day 2: Domain and Range
Day 3: Rates of Change and Graph Behavior
Day 4: Library of Parent Functions
Day 5: Transformations of Functions
Day 6: Transformations of Functions
Day 7: Even and Odd Functions
Day 8: Quiz 1.1 to 1.6
Day 9: Building Functions
Day 10: Compositions of Functions
Day 11: Inverse Functions
Day 12: Graphs of Inverse Functions
Day 13: Piecewise Functions
Day 14: Quiz 1.7 to 1.11
Day 15: Unit 1 Review
Day 16: Unit 1 Test
Unit 2: Polynomial and Rational Functions
Day 1: Connecting Quadratics
Day 2: Completing the Square
Day 3: Polynomials in the Short Run
Day 4: Polynomials in the Long Run
Day 5: Review 2.1-2.4
Day 6: Quiz 2.1 to 2.4
Day 7: Factor and Remainder Theorem
Day 8: Factor and Remainder Theorem
Day 9: Complex Zeros
Day 10: Connecting Zeros Across Multiple Representations
Day 11: Intro to Rational Functions
Day 12: Graphing Rational Functions
Day 13: Quiz 2.5 to 2.9
Day 14: Unit 2 Review
Day 15: Unit 2 Test
Unit 3: Exponential and Logarithmic Functions
Day 1: Exponential Functions
Day 2: Graphs of Exponential Functions
Day 3: Compound Interest and an Introduction to "e"
Day 4: Review 3.1-3.3
Day 5: Quiz 3.1 to 3.3
Day 6: Logarithmic Functions
Day 7: Graphs of Logarithmic Functions
Day 8: Logarithm Properties
Day 9: Solving Exponential and Logarithmic Equations
Day 10: Quiz 3.4 to 3.7
Day 11: Exponential and Logarithmic Modeling
Day 12: Unit 3 Review
Day 13: Unit 3 Test
Unit 4: Trigonometric Functions
Day 1: Right Triangle Trig
Day 2: Inverse Trig Ratios
Day 3: Radians and Degrees
Day 4: Unit Circle
Day 5: Unit Circle
Day 6: Other Trig Functions
Day 7: Review 4.1-4.6
Day 8: Quiz 4.1 to 4.6
Day 9: Graphing Sine and Cosine
Day 10: Transformations of Sine and Cosine Graphs
Day 11: Graphing Secant and Cosecant
Day 12: Graphing Tangent and Cotangent
Day 13: Quiz 4.7 to 4.10
Day 14: Inverse Trig Functions
Day 15: Trigonometric Modeling
Day 16: Trigonometric Identities
Day 17: Unit 4 Review
Day 18: Unit 4 Review
Day 19: Unit 4 Test
Unit 5: Applications of Trigonometry
Day 1: Law of Sines
Day 2: The Ambiguous Case (SSA)
Day 3: Law of Cosines
Day 4: Area and Applications of Laws
Day 5: Vectors
Day 6: Review 5.1-5.5
Day 7: Quiz 5.1 to 5.5
Day 8: Polar Coordinates
Day 9: Equations in Polar and Cartesian Form
Day 10: Polar Graphs Part 1
Day 11: Polar Graphs Part 2
Day 12: Review 5.6-5.9
Day 13: Quiz 5.6 to 5.9
Day 14: Parametric Equations
Day 15: Parametric Equations (With Trig)
Day 16: Unit 5 Review
Day 17: Unit 5 Test
Unit 6: Systems of Equations
Day 1: What is a Solution?
Day 2: Solving Systems with Substitution
Day 3: Solving Systems with Elimination
Day 4: Review 6.1-6.3
Day 5: Quiz 6.1 to 6.3
Day 6: Solving Systems in 3 Variables
Day 7: Solving Systems in 3 Variables
Day 8: Partial Fractions
Day 9: Unit 6 Review
Day 10: Unit 6 Test
Unit 7: Sequences and Series
Day 1: Introducing Sequences
Day 2: Using Sequences and Series to Describe Patterns
Day 3: Arithmetic Sequences and Series
Day 4: Review 7.1-7.2
Day 5: Quiz 7.1 to 7.2
Day 6: Geometric Sequences and Finite Series
Day 7: Infinite Geometric Sequences and Series
Day 8: Proof by Induction
Day 9: Proof by Induction
Day 10: Quiz 7.3 to 7.5
Day 11: Unit 7 Review
Day 12: Unit 7 Test
Unit 8: Limits
Day 1: What is a Limit?
Day 2: Evaluating Limits Graphically
Day 3: Evaluating Limits with Direct Substitution
Day 4: Evaluating Limits Analytically
Day 5: Evaluating Limits Analytically
Day 6: Review 8.1-8.4
Day 7: Quiz 8.1 to 8.4
Day 8: Continuity
Day 9: Continuity
Day 10: Intermediate Value Theorem
Day 11: Intermediate Value Theorem
Day 12: Review 8.5-8.6
Day 13: Quiz 8.5 to 8.6
Day 14: Limits at Infinity
Day 15: Unit 8 Review
Day 16: Unit 8 Test
Unit 9: Derivatives
Day 1: Introduction to Derivatives
Day 2: Average versus Instantaneous Rates of Change
Day 3: Calculating Instantaneous Rate of Change
Day 4: Calculating Instantaneous Rate of Change
Day 5: The Derivative Function
Day 6: The Derivative Function
Day 7: Review 9.1-9.3
Day 8: Quiz 9.1 to 9.3
Day 9: Derivative Shortcuts
Day 10: Differentiability
Day 11: Connecting f and f’
Day 12: Connecting f and f’
Day 13: Review 9.4-9.6
Day 14: Quiz 9.4 to 9.6
Day 15: Derivatives of Sine and Cosine
Day 16: Product Rule
Day 17: Quotient Rule
Day 18: Review 9.7-9.9
Day 19: Quiz 9.7 to 9.9
Day 20: Unit 9 Review
Day 21: Unit 9 Test
Unit 10: (Optional) Conic Sections
Day 1: Intro to Conic Sections
Day 2: Defining Parabolas
Day 3: Working with Parabolas
Day 4: Quiz 10.1 to 10.3
Day 5: Defining Ellipses
Day 6: Working with Elllipses
Day 7: Defining Hyperbolas
Day 8: Working with Hyperbolas
Day 9: Quiz 10.4 to 10.7
Day 10: Unit 10 Review
Day 11: Unit 10 Test
Learning Targets
Understand that the angles in a right triangle are determined by the ratio of the sides
Use the trigonometric ratios to find missing angles in a right triangle
Tasks/Activity | Time |
---|---|
Activity | 15 minutes |
Debrief Activity | 10 minutes |
Important Ideas | 5 minutes |
Check Your Understanding | 20 minutes |
Activity: What’s Your Angle?
Lesson Handouts
Media Locked
Media Locked
Answer Key
Media Locked
Homework
Media Locked
Experience First
This lesson was a collaborative effort with my good friend and colleague, Lindsey Gallas, from Stats Medic. Ever since we have both used this approach for teaching inverse trig to our students, we have found that students have a much better understanding of the conceptual underpinnings of this topic. The big idea of the activity is that whenever the sides of a right triangle are in a specific ratio, the angle is “set in stone”. Students look at three triangles where the opposite side is one-third of the hypotenuse. They reason that the missing angle must be 19.5˚ since all the triangles are similar and thus the angles must be congruent between triangles. The driving question becomes: “Is there a way to “look up” the angle when given a particular ratio?” The answer is, of course, inverse trigonometry! I make a big deal of pulling up tables like this one and tell students they can just pull out their handy dandy chart and figure out what angle goes with their given ratio. They are actually quite satisfied with this answer but I go on to tell them that their calculator actually has this whole table stored for them and then I proceed to tell them how they can access those tables on their calculator. The benefit of this approach is that students don’t lose sight of the fact that the angles in a right triangle are determined by the ratio of sides because of similar triangles. We want to avoid students skipping too quickly to procedures and memorized algorithms (“when I want to find an angle I use the sine with the little negative 1 next to it”)
Formalize Later
Although this lesson is very straightforward, use this as an opportunity to build good vocabulary and establish conceptual understanding of trigonometric ratios. These first two lessons of the unit are important stepping stones to understanding future ideas of the unit circle, special right triangles, and evaluating sine, cosine, and tangent for angles bigger than 90˚. As much as possible, we want to link ideas back to what they’ve learned in Geometry so we don’t shy away from terms like similar triangles, congruent angles, and proportional sides. A more formal lesson on inverse trig functions will come in lesson 4.7, so we avoid becoming overly technical about limited domains and ranges. Feel free to use informal language around what angle would have caused a particular ratio of sides instead of formal function notation.