Lesson #1

Lesson #1

Introduction to Trigonometry

Introduction to Math 7

• Who am I?• What is this class about?– Special types of functions, especially trigonometric ones– Different types of coordinates (versus (x, y))– Miscellany: Linear algebra, Sequences/Series, Conic

sections• What do you need to know for Math 7?– Algebra at the level of Math 6 (i.e. variables, functions,

logarithms, exponents, transformations)

Expectations• There is no written class contract• Respect for one another, without compromise• Complete (or near) silence when any one of us, teacher

or student, is speaking• Students will challenge the teacher and each other and

ask questions• The teacher will challenge the students and ask questions• Class participation is valued (literally)• ***Cheating and dishonesty is not tolerated – cite your

sources on homework and in class***

Grading

• Tests:• Homework:• Do Now/Class grades:• Extra credit?• This is a math class: proof and logic (a.k.a.

“showing your work” and “explaining your reasoning”) are critical at all times

Lesson Aims

• At the end of this lesson, students will:– Understand what trigonometry is– Understand some of the applications of

trigonometry– Be able to name the three major trigonometric

ratios

Trigonometry: What is it?

• A branch of mathematics that studies triangles, primarily (what are these?):

• Deals with relationships between the sides and the angles of triangles

• These relationships can be explained by trigonometric functions, like sine, cosine, and tangent

Trigonometric Functions• Three “basic” trigonometric functions (there are others)• Sine, the ratio of the side opposite an angle to the hypotenuse of a

right triangle• Cosine, the ratio of the side adjacent to an angle to the hypotenuse of

a right triangle• Tangent, the ratio of the side opposite an angle to the side adjacent to

an angle

Trigonometric functions• Sometimes math teachers talk about SOHCAHTOA, which is

an acronym for:

• sin(angle)=

• cos(angle)=

• tan(angle)=

• We will discuss these more in depth over the next few days…

Where did trigonometry come from?

• Word “trigonometry” derived from Greek word “trigonometria,” meaning “triangle measuring”

• But not just the Greeks studied trigonometry!• Egyptians used primitive trigonometry to construct pyramids• Indians focused on both theory and applications to astronomy (especially

why the Sun caused shadows to be cast in a certain way) and geography• By the 10th century, Arab and Persian mathematicians were using all of the

trigonometric functions with great accuracy– One of these Persian mathematicians was Muhammad ibn Musa al-Khwarizmi,

whose name inspired the word “algorithm” and whose use of the word “al-jabr” (“restoration of a broken bone”) in reference to solving equations inspired the word “algebra”

• Chinese and Western Europeans made further advances and rediscovered much in trigonometry up until about the 18th century

Why do we care about trigonometry today?

• Brainstorm with a partner: where might we care about the relationship of angles and lengths?

• “Short” List:– Construction– Architecture– Nasty things like catapults and cannons (unless used in the circus)– Sports (Pool, basketball, hockey)– Basketball– Waves (ocean, radio, light, etc.)!

• Graph sin(x) in your graphing calculators – what do you see?– Statistics– The stock market– Any place where ratios of things go up and down in some proportion?

Physics and Trigonometry

• This is a clip for a physics class, but can you spot the right triangle?

• http://www.youtube.com/watch?v=6rTN4-u2ibE

• Another clip showing where trigonometry can be used at the Basketball Hall of Fame:

• http://www.teachertube.com/viewVideo.php?video_id=1401&title=Trigonometry_at_the_Basketball_Hall_of_Fame

Closing Summary

• Trigonometry is the mathematics of triangles – the relationship between their angles and their sides

• We will see that trigonometry has applications beyond just triangles though…circles and waves, even

• Homework– Posted on board

Lesson #2

Introduction to Tangent

Lesson Aims

• Students will know when to use the tangent ratio to solve a problem relating to right triangles

• Students will be able to find the length of a side of a triangle using the tangent ratio

• Students will be able to find the measure of an angle of a triangle using the tangent ratio

• Students will be able to name a few applications of the tangent ratio

Do Now

• Examine the triangles on your table with a partner, using only a protractor and a ruler

• Measure every side of the triangles• Measure every angle of the triangles• What is different about the triangles?• What is the same?

Tangent

• The “TOA” portion of SOHCAHTOA

• Tan(angle) =

• Key thing to remember is this holds true for EVERY right triangle, so we can always use this ratio if given an opposite and adjacent side, or either side and the angle

Example of using tangent to solve for sides of a triangle

• A man uses a protractor to find his angle of elevation (from his feet) to the top of a tree of 45 degrees. (Draw this, teacher.) If he is 18 feet away from the base of the tree, how tall is the tree?

• By the way, how does this answer change if the man is 6 feet tall and is measuring his angle of elevation from approximately the top of his head?

Now you try• A woman looks down from the 102nd floor of the Empire State Building

and sees her friend standing 2 blocks away (~500 feet). Whipping out her protractor, she estimates that the angle of depression between her and her friend is 21.8 degrees. What is the approximate height of the Empire State Building up to the 102nd floor? (Teacher, draw this too!)

Example of using tangent to solve for the angle of a triangle

• Recall the concept of an “inverse function”• How can we “undo” tangent?

• A man stands 15 feet away from the base of Eliot Hall. If we know Eliot Hall is 40 feet high, and the man is ~6 feet tall, at what angle should he tilt his head to see the top of the building?

Now you try

• A right triangle, with its base being one leg, has height 20 cm and base 13 cm. What is the measure of the angle made by the hypotenuse and the base? (Draw this, teacher.)

Closing Remarks• All similar right triangles have the same ratio of any two

corresponding sides• An example of this is the tangent ratio, which is defined as

the length of the opposite side over the length of the adjacent side to an angle (TOA)

• To find either the opposite or adjacent side, given an angle and either the opposite or the adjacent side, use the tangent (tan) function

• To find an angle, given an opposite and an adjacent side, use the inverse tangent (tan-1 function)

• Homework: On board