2. History of Trigonometry
History of the Unit Circle
Important Triangles
SOH CAH TOA
Ratios
Quadrants
The Final Unit Circle
Fun Facts
Standards
Credits
3. Hipparchus of Nicaea is known as the father of
trigonometry. He compiled the first trigonometric
tables to simplify the study of astronomy more than
2000 years ago. He paved the way for other
mathematicians and astronomers using triangle ratios.
The term “trigonometry” itself emerged in the 16th
century, although it derives from ancient Greek roots:
“tri” (three), “gonos” (side), and “metros” (measure).
4. The idea of dividing a circle into 360 equal pieces dates
back to the sexagesimal (base 60) counting system of
the ancient Sumerians. The appeal of 60 was that it
was evenly divisible by so many numbers (2, 3, 4, 5, 6,
10, 12, 15, 20, and 30). Early astronomical calculations
wedded the sexagesimal system to circles, and the rest
is history.
6. Click here for a
SOHCAHTOA song.
SOH CAH TOA is an
acronym someone came
up with to help us
remember how to find
the sine, cosine, and
tangent values of an
angle. We use this in
forming the unit circle by
using the 30-60-90 and
45-45-90 triangles and
ratios.
Click here for
SOHCAHTOA
practice
7. Why was the number 60 used as the base for the degrees
in a circle?
A. It was Hipparchus’ favorite number
B. It is divisible by a lot of numbers
C. It was the year they invented trigonometry
9. Sorry, your answer is incorrect. Brush up on the history
slides and try again.
10. We’re going to use ratios and the two important
triangles to build the unit circle. First, we need to
remember the definition of ratios. A ratio is a quotient
of two numbers or quantities. Also, since we’re
building the UNIT circle, we need to remember that
“UNIT” means ONE. So we’re going to make each of
the hypotenuses of the important triangles equal to
one.
11. For the 30-60-90 triangle, we will need to divide each
side by 2 so that they hypotenuse will have a length of
1. Therefore, we’re left with a triangle that looks like
this:
12. For the 45-45-90 triangle, we will need to divide each
side by the square root of 2. We will then need to
rationalize each denominator and we’ll end up with a
triangle like this:
13. Once we have the triangles with 1 for the hypotenuse,
what is the side length opposite of 30°?
A.
B.
C.
15. Sorry, your answer is incorrect. Read the slide about the
ratios of important triangles again.
16. The unit circle is drawn on the coordinate plane so just
like the coordinate plane, we have four quadrants. Sine,
cosine and tangent are positive in exactly two quadrants
and negative in the other two quadrants. Sine
corresponds to the y-values and cosine corresponds to
the x-values. Tangent is a ratio of sine values to cosine
values.
Sine is positive in quadrants 1&2 and negative in 3&4
Cosine is positive in quadrants 1&4 and negative in 2&3
Tangent is positive in quadrants 1&3 and negative in 2&4
17. We can take the ratio versions of the 30-60-90 and 45-
45-90 triangles and place them inside a circle with
radius of one to create the final unit circle.
18. Trigonometry is everywhere in our lives even though
you may not have heard of it or know how to use the
sine, cosine or tangent functions.
For example the mathematics behind trigonometry is
the same mathematics that enables us to store sound
waves digitally on a CD. So when you’re burning your
favorite songs onto a CD, technically you’re using
trigonometry without even knowing it.
19. The sine and cosine wave (pictured below) are the
waves that are running through the electrical circuit
known as Alternating Current. So when you plug
something into the wall, which most of us do on a
daily basis, we are again using trigonometry.
20. 2.10.11.A – Identify, create, and solve practical problems involving right
triangles using the trigonometric functions and the Pythagorean Theorem.
2.1.G.C – Use ratio and proportion to model relationships between quantities.
1. Facilitate and Inspire Student Learning and Creativity
Teachers use their knowledge of subject matter, teaching and learning, and
technology to facilitate experiences that advance student learning, creativity,
and innovation in both face-to-face and virtual environments. Teachers:
a. promote, support, and model creative and innovative thinking and
inventiveness.
b. engage students in exploring real-world issues and solving authentic
problems using digital tools and resources.
c. promote student reflection using collaborative tools to reveal and clarify
students' conceptual understanding and thinking, planning, and creative
processes.
d. model collaborative knowledge construction by engaging in learning with
students, colleagues, and others in face-to-face and virtual environments.
