Session_1_Friday_1_16_2015

PRECALCULUS PREPARATIONSPRING SEMESTER – 2015 - PRECALCULUS

Course: Precalculus w/Trigonometry Delivered: Friday, January 16, 2015Instructor: Mr. Carlos Ortiz TBCB: Friday, January 16, 2015Student: ALAIN VILLASECA ETTC: 120-150 min

If at first you don’t succeed, then skydiving is not for you!

ANON

Alain, these identities were formulated many years ago during the time when the Greeks at the peak of their curiosities and they began to ask many questions about specific relationships. Let’s consider these two right triangles that contain some angles that should be quite familiar to us:

A 45o-45o-90o right triangle is simply half of a square.

If the original side of the square is a, then the diagonal of the square

(which would be the HYPOTENUSE Of the right triangle created when you cut it IN HALF) can be found by using the Pythagorean Theorem:

Since h represents the SIDE OF A TRIANGLE, we ignore the negative answer and so we have:

BIG DEAL…SO WHAT?

A 30o-60o-90o

right triangle is a bit different.

If the shorter LEG is a, then the longer leg is will be a MULTIPLIED by and the hypotenuse will be TWICE a: 2a. NOW, we didn’t find these using the Pythagorean Theorem, but we can verify them:

BIG DEAL…SO WHAT?

PRECALCULUS PREPARATION – 1/16/2015 TOPIC = TRIGONOMETRIC FUNCTIONSMr. Carlos Ortiz | ALAIN VILLASECA page 1 of 9

Copyright © 2015 Carlos Ortiz

Well, it turns out that ANY RIGHT TRIANGLE that has TWO LEGS THAT ARE THE SAME, will ALWAYS HAVE A HYPOTENUSE THAT IS TIMES LONGER!

AND

ANY RIGHT TRIANGLE that has ONE LEG THAT IS LONGER THAN THE OTHER, will ALWAYS HAVE A HYPOTENUSE THAT IS TIMES LONGER!

AND SO THE GREEKS figured that if this was true, then certain RATIOS would ALWAYS HAVE THE SAME VALUE:

EXAMPLE 1: For the 45o-45o-90o right triangle, the ratio of the SIDE OPPOSITE THE 45o angle to the hypotenuse would ALWAYS BE be:

It turns out that specifically for 45o, the ratio of the SIDE ADJACENT THE 45o angle to the hypotenuse would ALWAYS BE be:

CONCLUSION: They decided to call the ratio of OPPOSITE OVER HYPOTENUSE the “sin” of the angle and the ratio of the ADJACENT OVER HYPOTENUSE would be called the “cos” of the angle.



AND

PRACTICE

P1 A right triangle has a hypotenuse of length 17 m and the shorter leg is 8 m long. What is the length of the longer leg?

a. 64 mb.

c.d. 225 m

P2 Consider a 45o-45o-90o right triangle. What is the ratio of the hypotenuse to either leg?

a. 1b. 0.707c. 2d. 1.414

P3 In a right triangle that contains a 60o angle and a shorter leg length of 2 m, what is the length of the hypotenuse?

a. 4 mb. 2 mc. 3.464 md. 1 m

P4 Consider the work the Greeks completed with the 45o-45o-90o right triangle. If they

had done similar work with the 30o-60o-90o right triangle, what sin and cos values would they have calculated for

a) 30o ?For sin: For cos:

b) 60o ?For sin: For cos:



CONCLUSION: The sin and cos values for 30o, 45o, and 60o WILL ALWAYS BE THE SAME, EVEN IF THEY ARE NOT PART OF A RIGHT TRIANGLE (this can be confirmed with any calculator).

An angle can be represented in math with ANY VARIABLE and GREEK VARIABLES tend to be pretty popular. If we call our ANGLE theta (here is that Greek letter: ϴ), then we can SUMMARIZE our results so far with this table:

ϴ = 30o ϴ = 45o ϴ = 60o

sin ϴ

cos ϴ

ALAIN, notice anything interesting about the values in this table?



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Session_1_Friday_1_16_2015

Documents

o right triangle

o angle

o sin cos alain

hypotenuse of length

practice p1a right triangle

right triangles

longer leg

shorter leg length