finding exact trigonometric values instructor brian d. ray
TRANSCRIPT
FINDING EXACT TRIGONOMETRIC VALUES
Instructor Brian D. Ray
DRILL• DIRECTIONS: Solve each special right
triangle shown below.
1)1
X
y
45
1
S t60
= 1
= 2 = 2= 3
2)
• 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.
2
DRILL• DIRECTIONS: Solve each special right
triangle shown below.1)
1
x
y45
1
S t60
= 1
= 2 = 2= 3
2)
• In the 30 – 60 – 90 triangle, assume that the short leg is 1.• How do we know which leg is the short leg?
The short leg is opposite the angle.30• The hypotenuse is 2 units according to the derivation we did in our previous unit.
• The hypotenuse is units long by the Pythagorean Theorem.3
OUR ULTIMATE GOAL
• Do you remember what kind of function we used to model each situation?
Time (in hrs)
0.5 1 1.5 2
Distance (miles)
30 60 90 120
• Why do we learn about functions?
OUR ULTIMATE GOAL
• Do you remember what kind of function we used to model each situation?
Ground zero
Path of baseball
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 7:28 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?
HERE’S THE POINT• Have you ever seen this before?
• What about these?
Let’s look here: http://www.truveo.com/How-to-make-a-y
oyo-sleep-Sleeper-yoyo-trick/id/2310084845
• What function do we have to model this motion?
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.
EXAMPLE
• Find the six trigonometric values for .
240
Step 1. Draw the angle.
90
180
270
360
Step 2. Find the reference angle.
60
Step 3. Set up the special right triangle. Be careful to use the correct signs.
3
1
2
Step 4. Apply the definitions we learned from the reference angle to find the trigonometric values.
sin240 opposite
hypotenuse
hypotenuse
adjacent240cos
adjacent
opposite240tan
oppositeadjacent
1
3
3
2
1
2
3
1
3
EXAMPLE
• Find the six trigonometric values for .
240
Step 1. Draw the angle.
90
180
270
360
Step 2. Find the reference angle.
60
Step 3. Set up the special right triangle. Be careful to use the correct signs.
3
1
2
Step 4. Apply the definitions we learned from the reference angle to find the trigonometric values.
csc240 hypotenuse
opposite
sec240 hypotenuse
adjacent opposite
adjacent240cot
oppositeadjacent
1
3
2
3
2
1
1
3
3
3
2 3
3
2
EXAMPLE 2
• Find the six trigonometric values for .
54
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.
1
Step 4. Apply the definitions we learned from the reference angle to find the trigonometric values.
oppositeadjacent
1
44
64
32
84
2
4
1
2
1
hypotenuse
opposite
4
5sin
hypotenuse
adjacent
4
5cos
adjacent
opposite
4
5tan
2
1
1
1
1
422
2
2
2
1
2
2
EXAMPLE 2
• Find the six trigonometric values for .
54
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.
1
Step 4. Apply the definitions we learned from the reference angle to find the trigonometric values.
oppositeadjacent
1
44
64
32
84
2
4
1
2
1
422
opposite
hypotenuse
4
5csc
adjacent
hypotenuse
4
5sec
opposite
adjacent
4
5cot
1
2
1
2
1
1 1
2
2
EXAMPLE
• Find the six trigonometric values for .330
Step 1. Draw the angle.
90
180
270
360
Step 2. Find the reference angle.Step 3. Set up the special right
triangle. Be careful to use the correct signs.
3
1
2
Step 4. Apply the definitions we learned from the reference angle to find the trigonometric values.
hypotenuse
opposite330sin
hypotenuse
opposite330cos
hypotenuse
opposite330tan
oppositeadjacent 3
1
2
1
2
3
3
1
3
3
30
EXAMPLE
• Find the six trigonometric values for .330
Step 1. Draw the angle.
90
180
270
360
Step 2. Find the reference angle.Step 3. Set up the special right
triangle. Be careful to use the correct signs.
3
1
2
Step 4. Apply the definitions we learned from the reference angle to find the trigonometric values. opposite
adjacent 3
1
30
opposite
hypotenuse330csc
adjacent
hypotenuse330sec
opposite
adjacent330cot
1
2
3
2
1
3
3
2
3
32
Quadrantal Angles
• Definition. A quadrantile angle is an angle whose initial side lies on one of the coordinates axes.
• Examples.
90
270
• How do we find trig values in this case?
90
180
270
360
90
180
270
360
Trigonometric Values of Quadrantal Angles
• Definition. The unit circle is a circle whose radius is 1 unit long.
90
180
270
360
1
( , )( , )
( , )( , )
0110
0110
• Identify the ordered pair for each quadrantal angle.
• We will now find out how to find calculate the trigonometric values of these angles.
EXAMPLE: Quadrantal Angles
90
180
270
360
1
( , )( , )
( , )( , )
0110
0110
• Find the six trigonometric values for .180
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.
r
y180sin
1
0 0
r
x180cos
1
1 1
x
y180tan
1
0
0
EXAMPLE: Quadrantal Angles
90
180
270
360
1
( , )( , )
( , )( , )
0110
0110
• Find the six trigonometric values for .180
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.
y
r180csc
1
0 0
x
r180sec
1
1
1
y
x180cot
0
1 undefined
Quadrantal AnglesTry This
90
180
270
360
1
( , )( , )
( , )( , )
0110
0110
• Find the six trigonometric values for .270
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.
r
y 270sin
1
1 1
r
x 270cos
1
0 0
x
y 270tan
0
1 undefined
Quadrantal AnglesTry This
90
180
270
360
1
( , )( , )
( , )( , )
0110
0110
• Find the six trigonometric values for .270
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.
y
r 270csc
1
1 1
x
r 270sec
0
1
0 y
x 270cot
1
0
undefined