Geometry
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Lesson 3.1 INTRODUCTION TO TRIGONOMETRY Objective: Students will be able to:- identify the hypotenuse in a right angled triangle, and the opposite and adjacent sides of a given angle - solve problems including those involving angles of elevation and depression using trigonometry find sine, cosine, and tangent of each acute angle given the side lengths of a right triangle with acute angles, -
Standards: G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G.SRT.6 Using a corresponding angle of similar right triangles, show that the relationships of the side ratios are the same, which leads to the definition of trigonometric ratios for acute angles. G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems G.SRT.8 Apply both trigonometric ratios and Pythagorean Theorem to solve application problems involving right triangles. Enduring Understandings 1. Certain right triangles have properties that allow their side lengths to be determined without using the Pythagorean Theorem. 2. Ratios can be used to find side lengths and angle measures of a right triangle when certain combinations of side lengths and angles measures are known. 3. The angles of elevation and depression are the acute angles of right triangles formed by a horizontal distance and a vertical height. 4) the Pythagorean Theorem and trigonometric ratios can be used to solve right triangles. 5) there is a relationship between sine and cosine of complementary angles. 6) Trigonometry can help us solve real world problems that involve triangles. 7) Use the basic trigonometric functions to solve for missing parts of right triangles Essential Questions
1. How are the properties of similar triangles used to create trigonometric ratios? 2. What are the relationships between the sides and angles of right triangles? 3. On what variable inputs are the ratios in trigonometry dependent? 4. What is the relationship between the sine and cosine ratios for the two acute angles in a right triangle? 5. How do trigonometric ratios relate to similar right triangles? 6) How are the trigonometric ratios useful in modeling real life situations? 7) How can the properties of right triangles and trigonometry be used to solve mathematical and real world problems? Teacher:Dr. Elie Tcheimegni
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Geometry
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Lesson 3.1 INTRODUCTION TO TRIGONOMETRY
Prior Knowledge Students should have knowledge of the concept of ratio and should know that division is not commutative. Students should be able to use a calculator to convert fractions, correct to three decimal places, measure the lengths of sides of a triangle to the nearest millimeter and draw to scale. Students should have studied Pythagoras’ Theorem and know the meaning of the term “hypotenuse”. Students may have studied similar triangles in geometry – the fact that corresponding sides are proportional makes trigonometry possible. Students will have understood how to calculate mean. Students should know how to use a measuring tape and protractor.
Introducing the Topic Trigonometry is simply geometrical constructions where the ratio of the side lengths in triangles is used to determine angular measurement. Trigon, meaning triangle and metria, meaning measurement Mathematicians have used trigonometry for centuries to accurately determine distances without having to physically measure them . It can also be used to calculate angles that would be very difficult to measure. Trigonometry
Videos related to the lesson INTRODUCTION TO TRIGONOMETRY PART I https://www.youtube.com/watch?v=zBI9Dh46p6I INTRODUCTION TO TRIGONOMETRY PART II https://youtu.be/__cPihQjqyI INTRODUCTION TO TRIGONOMETRY PART III https://youtu.be/qvP5npWC7sU
Resources Required Calculators, measuring tapes, graph paper, cm rulers and a spray or sunny day!
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Geometry
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Lesson 3.1 INTRODUCTION TO TRIGONOMETRY EXERCISE I:Identifying Opposite, Adjacent and Hypotenuse
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Geometry
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Lesson 3.1 INTRODUCTION TO TRIGONOMETRY EXERCISE II:Writing Sine, Cosine, Tangent Ratios
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Geometry
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Lesson 3.1 INTRODUCTION TO TRIGONOMETRY EXERCISE III 1)
2)
3)
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Geometry
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Lesson 3.1 INTRODUCTION TO TRIGONOMETRY 4)
5)
6)
7)
8)
9)
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Geometry
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Lesson 3.1 INTRODUCTION TO TRIGONOMETRY
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Geometry
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Lesson 3.1 INTRODUCTION TO TRIGONOMETRY
3)
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Geometry
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Lesson 3.1 INTRODUCTION TO TRIGONOMETRY EXERCISE IV
Find the value of each given trigonometric function using the Pythagorean identity.
d)
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F)
G)
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Geometry
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Lesson 3.1 INTRODUCTION TO TRIGONOMETRY EXERCISE V 1) Given the triangle at the right, express the exact value of the six trig functions in relation to theta.
B) A)
2)
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Geometry
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Lesson 3.1 INTRODUCTION TO TRIGONOMETRY EXERCISE VI:Find x. Round your answers to the nearest tenth
5) 4)
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Geometry
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Lesson 3.1 INTRODUCTION TO TRIGONOMETRY PROBLEM I: The goal of this problem is to find angle
in the figure below
1) Consider triangle HRI Adjacent to angle Opposite to angle =_____________________ Tangent =__________________
=________________
in degree =
2) Using supplementary angle theorem, find the value of gamma (
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Geometry
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Lesson 3.1 INTRODUCTION TO TRIGONOMETRY 3)Find the value of beta (
)
in degree
In triangle CRI Opposite side to beta =___________________ Tangent beta =_________________________
Adjacent side to
=_______________
=____________________ =________________________
4) Find the value of
Teacher:Dr. Elie Tcheimegni
in the figure below
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Geometry
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Lesson 3.1 INTRODUCTION TO TRIGONOMETRY
Magic Hexagon for Trig Identities
1) Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression.
2) Use the definition or identities to find the exact value of each of the remaining five trigonometric functions of the acute angle θ.
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Geometry
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Lesson 3.1 INTRODUCTION TO TRIGONOMETRY 3)
4) Write the expression tan 265º as the function of an acute angle of measure less than 45º. 5)
6)
7)
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Geometry
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Lesson 3.1 INTRODUCTION TO TRIGONOMETRY 1)
2)
3)
4)
Teacher:Dr. Elie Tcheimegni
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Geometry
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Lesson 3.1 INTRODUCTION TO TRIGONOMETRY 5)
6)
7)
8)
9)
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Geometry
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Lesson 3.1 INTRODUCTION TO TRIGONOMETRY 10)
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