Curriculum / Math / 11th Grade / Unit 7: Trigonometric Identities and Equations / Lesson 8
Math
Unit 7
11th Grade
Lesson 8 of 16
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Use inverse trigonometric functions to solve contextual problems.
The core standards covered in this lesson
F.TF.B.7 — Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
The foundational standards covered in this lesson
F.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
Suggestions for teachers to help them teach this lesson
Many of the problems presented are climate related, but it is important for students to see other kinds of functions besides these.Â
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
25-30 minutes
The average monthly temperature, in degrees Celsius, in a coastal city in the United States is periodic and can be modeled with the equation $${y=-8\mathrm{cos}\left((x-1)\left(\pi\over6\right)\right)+17.5}$$, where $$x$$ represents the month, $$x=1$$ represents January, and $$y$$ represents the average temperature in degrees Celsius.
Below is the graph of the equation.
A tourist wants to visit the city when the average temperature is closest to 25℃. What recommendations would you make regarding when the tourist should travel?
Precalculus and Advanced Topics > Module 4 > Topic C > Lesson 14 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..
Do the activity Burning Daylight by Desmos. Skip slides 6 and 7.
Burning Daylight by Desmos is made available by Desmos. Copyright © 2017 Desmos, Inc. Accessed May 21, 2018, 2:21 p.m..
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
5-10 minutes
The estimated size for a population of rabbits and a population of coyotes in a desert habitat are shown in the table. The estimated population sizes were recorded as a part of a long-term study related to the effect of commercial development on native animal species.
Below are sketches of each graph, with a curve fitted to the data.
Write the function for each situation and then calculate the two different numbers of years where the population estimate is about 13,000 for rabbits and 2,100 for coyotes.
The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.
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Solve quadratic trigonometric equations.
Topic A: Basic Trigonometric Identities and Equivalent Expressions
Derive and verify trigonometric identities using transformations and equivalence of functions.
Standards
F.TF.C.8
Derive and use the Pythagorean identity to write equivalent expressions.
Verify trigonometric identities using Pythagorean and reciprocal identities.
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Topic B: Solve Trigonometric Equations
Find angle measures using inverse trig functions in right triangles.
F.TF.B.6F.TF.B.7
Analyze inverse trigonometric functions graphically.
F.BF.B.4.DF.IF.C.7.EF.TF.B.6F.TF.B.7
Solve linear trigonometric equations.
F.TF.B.7
Solve linear trigonometric equations using $$u$$-substitution.
Solve trigonometric equations using identities.
F.TF.B.7F.TF.C.8
Topic C: Advanced Identities and Solving Trigonometric Equations
Evaluate expressions using sum and difference formulas.
F.TF.C.9
Solve equations and prove identities using sum and difference formulas.
Derive double angle formulas and use them to solve equations and prove identities.
Use trigonometric identities to analyze graphs of functions.
F.TF.C.8F.TF.C.9
Topic D: Applications and Extensions of Trigonometric Functions
Use the Law of Sines to find missing side lengths and angle measures in acute triangles.
G.SRT.D.10
Find missing side lengths and angle measures using the Law of Cosines in acute triangles.
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