FINDING EXACT TRIGONOMETRIC VALUES Instructor Brian D. Ray.

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


Ground zero

Path of baseball

OUR ULTIMATE GOAL


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


240


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


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


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


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


54

Step 1. Draw the angle.Step 2. Find the reference angle.Step 3. Set up the special right


1


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


90

180

270

360

Step 2. Find the reference angle.Step 3. Set up the special right


3

1

2


hypotenuse

opposite330sin

hypotenuse

opposite330cos

hypotenuse

opposite330tan

oppositeadjacent 3

1

2

1

2

3

3

1

3

3

30

EXAMPLE



90

180

270

360

Step 2. Find the reference angle.Step 3. Set up the special right


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


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





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





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





y

r 270csc

1

1 1

x

r 270sec

0

1

0 y

x 270cot

1

0

undefined

FINDING EXACT TRIGONOMETRIC VALUES Instructor Brian D. Ray.

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