FINDING EXACT TRIGONOMETRIC VALUES

1
FINDING EXACT TRIGONOMETRIC VALUES

• Instructor Brian D. Ray

2
DRILL

• DIRECTIONS Solve each special right triangle
  shown below.

1)
2)
y
1
S
t
2
X
1
1
What I NEED To Remember

• In the 45 45 90 triangle, assume that a leg
  is 1.

• The other leg is 1 since the 45 45 90 is
  isosceles!

• The hypotenuse, by the Pythagorean Theorem is
  units long.

3
DRILL

• DIRECTIONS Solve each special right triangle
  shown below.

1)
2)
y
1
S
t
2
x
1
1
What I NEED To Remember

• In the 30 60 90 triangle, assume that the
  short leg is 1.

• How do we know which leg is the short leg?

• The hypotenuse is 2 units according to the
  derivation we did in our previous unit.

4
OUR ULTIMATE GOAL

• Why do we learn about functions?

• Do you remember what kind of function we used to
  model each situation?

5
OUR ULTIMATE GOAL
Path of baseball

• Do you remember what kind of function we used to
  model each situation?

Ground zero
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OUR ULTIMATE GOAL

• Do you remember what kind of function we used to
  model each situation?

Verizon charges me 0.45 for each additional
minute that I use beyond my plan. I used 728
additional minutes, but of course, Verizon will
round up, rather than round down. What function
can I use to model this the additional cost I
would pay?
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HERES THE POINT

• Have you ever seen this before?

Lets look here http//www.truveo.com/How-to-mak
e-a-yoyo-sleep-Sleeper-yoyo-trick/id/2310084845

• What about these?

• What function do we have to model this motion?

8
OBJECTIVE

• To model the situations given in the last slides,
  we need to learn more trigonometry! Our
  objective is to calculate the trigonometric value
  of any angle, particularly those having special
  reference angles.

9
EXAMPLE

• Find the six trigonometric values for .

Step 1. Draw the angle.
Step 2. Find the reference angle.
Step 3. Set up the special right triangle. Be
careful to use the correct signs.
Step 4. Apply the definitions we learned from
the reference angle to find the trigonometric
values.
opposite
adjacent
10
EXAMPLE

• Find the six trigonometric values for .

Step 1. Draw the angle.
Step 2. Find the reference angle.
Step 3. Set up the special right triangle. Be
careful to use the correct signs.
Step 4. Apply the definitions we learned from
the reference angle to find the trigonometric
values.
opposite
adjacent
11
EXAMPLE 2

• Find the six trigonometric values for .

Step 1. Draw the angle.
Step 2. Find the reference angle.
Step 3. Set up the special right triangle. Be
careful to use the correct signs.
Step 4. Apply the definitions we learned from
the reference angle to find the trigonometric
values.
opposite
adjacent
12
EXAMPLE 2

• Find the six trigonometric values for .

Step 1. Draw the angle.
Step 2. Find the reference angle.
Step 3. Set up the special right triangle. Be
careful to use the correct signs.
Step 4. Apply the definitions we learned from
the reference angle to find the trigonometric
values.
opposite
adjacent
13
EXAMPLE

• Find the six trigonometric values for .

Step 1. Draw the angle.
Step 2. Find the reference angle.
Step 3. Set up the special right triangle. Be
careful to use the correct signs.
Step 4. Apply the definitions we learned from
the reference angle to find the trigonometric
values.
opposite
adjacent
14
EXAMPLE

• Find the six trigonometric values for .

Step 1. Draw the angle.
Step 2. Find the reference angle.
Step 3. Set up the special right triangle. Be
careful to use the correct signs.
Step 4. Apply the definitions we learned from
the reference angle to find the trigonometric
values.
opposite
adjacent
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Quadrantal Angles

• Definition. A quadrantile angle is an angle
  whose initial side lies on one of the coordinates
  axes.

• Examples.

• How do we find trig values in this case?

The Unit Circle!
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Trigonometric Values of Quadrantal Angles

• Definition. The unit circle is a circle whose
  radius is 1 unit long.

( , )
( , )

• Identify the ordered pair for each quadrantal
  angle.

( , )
( , )
The Unit Circle!

• We will now find out how to find calculate the
  trigonometric values of these angles.

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EXAMPLE Quadrantal Angles
( , )

• Find the six trigonometric values for .

( , )
Step 1. Draw the angle.
Step 2. Find the ordered pair from the unit
circle..
( , )
Step 3. Apply the definitions we learned from
the reference angle to find the trigonometric
values.
( , )
The Unit Circle!
18
EXAMPLE Quadrantal Angles
( , )

• Find the six trigonometric values for .

( , )
Step 1. Draw the angle.
Step 2. Find the ordered pair from the unit
circle..
( , )
Step 3. Apply the definitions we learned from
the reference angle to find the trigonometric
values.
( , )
The Unit Circle!
19
Quadrantal AnglesTry This
( , )

• Find the six trigonometric values for .

( , )
Step 1. Draw the angle.
Step 2. Find the ordered pair from the unit
circle..
( , )
Step 3. Apply the definitions we learned from
the reference angle to find the trigonometric
values.
( , )
The Unit Circle!
20
Quadrantal AnglesTry This
( , )

• Find the six trigonometric values for .

( , )
Step 1. Draw the angle.
Step 2. Find the ordered pair from the unit
circle..
( , )
Step 3. Apply the definitions we learned from
the reference angle to find the trigonometric
values.
( , )
The Unit Circle!