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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
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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 alKhwarizmi, 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:
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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=6rTN4u2ibE
• 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