TRIGONOMETRY Math 10 Ms. Albarico

LESSONS sequence Introduction to Trigonometry Review on Types of Triangles Labelling Triangles Trigonometric Ratios Solving Word Problems Related to Trigonometry

Introduction Trigonometry is a branch of mathematics that uses triangles to help you solve problems. Trigonometry is useful to surveyors, engineers, navigators, and machinists (and others too.)

Triangles Around Us

Math 10 Trigonometry Opening Activity DESIGNING TRIANGLES Math 10 Trigonometry Opening Activity

Designing Triangles For this activity, you'll need: a piece of paper, pencil a ruler, protractor some coloured pencils or pens

Designing Triangles What To Do: 1) Using your pencil and ruler, draw some straight lines on your piece of paper to make an interesting pattern. You can draw as many or as few as you like. 2) Using your coloured pencils or pens, decorate all the three-sided shapes in some way. You could colour them all in using a particular colour or you could cover them with a special design or pattern

When you finish your work, you have to explain your design by answering this questions: What is your pattern all about? Can you describe what you see in your own pattern? Can you find any shapes which have three sides? How about any with four sides? Which shape or shapes have the least sides? What are the measures of their angles? What is the total sum of their angles?  

EXPECTED OUTCOMES

Students are expected to: M04.01 Explain the relationships between similar right triangles and the definitions of the primary trigonometric ratios. M04.02 Identify the hypotenuse of a right triangle and the opposite and adjacent sides for a given a cute angle in the triangle. M04.03 Solve right triangles, with or without technology. M04.04 Solve a problem that involves one or more right triangles by applying the primary trigonometric ratios or the Pythagorean theorem. M04.05 Solve a problem that involves indirect and direct measurement, using the trigonometric ratios, the Pythagorean theorem, and measurement instruments such as a clinometer or metre stick.

Vocabulary perpendicular parallel sides angle triangle congruent similar dilate sail navigate approach

Key Terms Primary Trigometric Ratios Sine Ratio Cosine Ratio Tangent Ratio Pythagorean Theorem Angle of Inclination Angle of Elevation Angle of Depression

What is a TRIANGLE Brainstorm students. They should come up with key words to describe TRIANGLE.

Triangles classified according to their Sides: Scalene Triangle. A scalene triangle that has no equal sides. Isosceles Triangle. An isosceles triangle is a triangle that has two equal sides. Equilateral Triangle. An equilateral triangle is a triangle that has three equal sides.

Identify the following triangles: Triangle B Triangle A Triangle C

Triangles classified according to their Angles: Right Triangle. A right triangle has a 900 angle. Obtuse Triangle. An obtuse triangle has one angle that has bigger than 900 angle. Acute Triangle. In an acute triangle, all angles has less than 900 angle.

Identify the following triangles: Triangle B Triangle A Triangle C

INTERACTIVE TRIANGLE

Labeling Right Triangles Names of the sides: the HYPOTENUSE the OPPOSITE side The ADJACENT side

Labeling Right Triangles A REFERENCE ANGLE must be selected before you start labeling the sides of the right angle It can be either of the two acute angles. Draw in the white board an right angled triangle so you can let the students decide which is there reference angle and work from there.

Labeling Right Triangles The HYPOTENUSE is the longest side of the right triangle. Because it is always found across from the right angle.

Labeling Right Triangles The OPPOSITE SIDE is the side across the 900 angle. Based from our drawing, which is our opposite side?

Labeling Right Triangles The ADJACENT SIDE is the side next to our reference angle.

Labeling Right Triangles The ADJACENT SIDE is the side next to our reference angle. Because it is always found across from the right angle.

Labeling Right Triangles Given the same triangle, how would the sides be labeled if we choose the other acute angle as our reference angle? Will there be any difference? Which side does not change? Answer - It changes the adjacent and opposite sides but not the hypotenuse. Change the reference angle to prove your point.

Key Terms: Primary Trigonometric Ratios – constant values based on the ratios of sides for particular angles in right-angled triangles. Sine, Cosine, and Tangent are called primary trigonometric ratios. tan X – a constant value based on the ratio of the length of the side to a chosen angle X in a right triangle. sin X – a constant value based on the ratio of the length of the side opposite to a chosen angle X to the length of the hypotenuse in a right triangle. cos X – a constant value based on the ratio of the length of the side adjacent to a chosen angle X to the length of the hypotenuse in a right triangle. Angle X is the reference angle.

5 3 Note: You can pick one triangle that you like from the Opening Activity. Identify the adjacent and opposite sides and your reference angle. x 4

Primary Trigonometric Ratios 3 5 x 4

Primary Trigonometric Ratios 5 3 x 4

Trigonometric Ratios S O H C A H T O A Take the first letter of each word. S O H C A H T O A

Note: Given Looking for Use Ratio of sides Angle measure SIN-1 COS-1 TAN-1 Angle, side Missing side SIN, COS, TAN Calculator Commands Reminder Set your calculator to ‘Degree’….. MODE (next to 2nd button) Degree (third line down… highlight it) 2nd Quit

Calculator Commands Set your calculator to ‘Degree’….. MODE (next to 2nd button) Degree (third line down… highlight it) 2nd Quit

Let’s Practice…. B c a C b A Let’s switch angles! Write the ratio for sin A Sin A = a c Write the ratio for cos A Cos A = b Write the ratio for tan A Tan A = a b B c a C b A Students should provide the ratios for REFERENCE ANGLE B and check them. Let’s switch angles! Find the sin, cos and tan for Angle B.

3 5 SOLVING FOR ANGLE X x 4 Ask the students – What did you notice with the angles? No matter what trigonometric ratios that you will use, the angle must be the same.

To solve for Angles: A B C xo Opp’ Adj’ Now we need to look at the two ratios involving the hypotenuse: hyp’ Opposite sin xo = Hypotenuse Adjacent cos xo = Hypotenuse

Calculator Commands For Trigonometric Inverse Functions: Press 2nd, use SIN for SIN-1 COS for COS-1 TAN for TAN-1

Trigonometric Ratios Sine Cosine Tangent Sin Cos Tan Name “say” Sine Cosine Tangent Abbreviation Abbrev. Sin Cos Tan Ratio of an angle measure sinθ = opposite side hypotenuse cosθ = adjacent side tanθ =opposite side adjacent side

Questions! What are the three sides of a right angle triangle? In your calculator, what function will we use to find angle measures? How can you remember easily the trigonometric ratios?

Open your textbook to page 70.

Assignment 1

Tangent Ratio

QUICK Review…. C 3cm B 4cm A Process: Find an angle that has a tangent (ratio) of 3 4 Round your answer to the nearest hundredth degree. C 3cm B 4cm A Process: I want to find an ANGLE. I was given the sides (ratio). Tangent = opposite adjacent Solution: TAN-1(3/4) = 36.87° Ask the student – What is your reference angle?

CLASS BOARD WORK

INDIVIDUAL CLASS WORK #10-14 ON page 76.

HOMEWORK #15-19 ON page 75-76.

ASSIGNMENT 2 Reference: Page 77 of (PEARSON CANADA) Foundations and Pre-Calculus Math 10

ASSIGNMENT 2

ASSIGNMENT 2 Let the students color their work.

USING TANGENT RATIO TO CALCULATE LENGTHS

CLASS BOARD WORK

INDIVIDUAL CLASS WORK #3-8 ON page 82.

HOMEWORK #10-16 ON page 83.

ASSIGNMENT 3 By partner, bring the following: Piece of string Heavy object (stopper) Needle Drinking straw (must be at least 15cm long) Read pages 85-86.

How are you going to measure an inaccessible height? Brainstorm students. Ask the question.

ASSIGNMENT 3 1) Students will create their own clinometer by pair. 2) Teacher will assign which inaccessible height to measure. 3) Students answer E-G on page 86 and Assess Your Understanding #1-3

Resources: (PEARSON CANADA) Foundations and Pre-Calculus Math 10 (NELSON CANADA) Mathematical Modelling Book 1 Polyhedron figures

For Tutorial http://www.sd43.bc.ca/Resources/ParentResources/math/10/Pages/Trigonometry.aspx https://www.mathsisfun.com/rightangle.html

References: https://www.mathsisfun.com/geometry/triangles-interactive.html http://www.math10.ca/lessons/measurement/trigonometryOne/trigonometryOne.php mathworld.wolfram.com