i. introduction (from geometry) ii. word problems and alt trig functions

I. Introduction (from Geometry)II. Word Problems and alt trig functions

III. Reference AnglesIV. Area and Heron’s FormulaV. The Law of SinesVI. The Law of Sines Part 2: The Ambiguous Case VII. Law of Cosines

Law of Sines and Cosines practice

VIII. RadiansIX. Unit Circle

OPENERSREVIEWSHOMEWORK

TRIGONOMETRY

TRIGONOMETRY

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OPENERS

ABCDEFGHIJKLMNOPQRST

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Introduction to Trigonometry

presentation

(from Geometry)

MAIN MENUPart 1: Opposite, Adjacent & HypotenusePart 2: Using trig functions on a calculatorPart 3: The consistent ratios

(activity with worksheet)Part 4: Trig definitionsPart 5: Solving for sides of a trianglePart 6: Solving for angles of a triangle

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ACT MENU

Part 7: Applications of Trig

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1. As a plane flies over East Leyden (2.5 miles away), The angle of elevation to the plane is 720. How high up is the plane?

2. Godzilla is 400 ft tall. If the angle of elevation to the top of his head is 480 from where you stand, how far away is Godzilla?

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3. Superman is flying at an altitude of 1,000 ft. He looks and sees Dr. Evil at an angle of declension (angle of depression) of 730. How far is Superman from Dr. Evil?

4. If the angle of elevation to the sun is 200, and a man casts a 7ft shadow, How tall is he (in feet and inches?).

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5. Jamie flies a kite at an angle of elevation of 480. If she has let out 60 ft of string, how high is the kite?

6. A statue of Mickey Mouse is at the top of a cliff. Bob, standing 100 feet from the cliff notes the angle of elevation to the top of the cliff is 370, and the angle of elevation to the top of the statue is 430. How tall is the statue?

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7. A giant bird lands on the top of a telephone pole. You are standing 10 feet from the bottom of the pole, and the angle of elevation to the top of the pole is 670. The angle of elevation to the top of the bird is 710. How tall is the bird?

8. From the top of your house, the angle of declension to the close side of the street is 800. The angle of declension to the

far side of the street is 760. If your house is 20 ft high, how wide is the street?

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Reference angles

Draw a 10 degree angle

x

y

10

We always start from this spot.

The positive x axis.

We always go the same direction.

Counterclockwise.

Reference Angles

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Reference angles


x

y

20

Reference Angles

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Reference angles


x

y

80

Reference Angles

TRIGONOMETRY

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Reference angles


x

y

120

Reference Angles

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Reference angles


x

y

315

Reference Angles

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Reference angles


x

y

45

Now we are going to use this drawing to find the values of sin, cos and tan for 45 degrees

Reference Angles

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x45

Reference Angles

1. Connect the end of your arrow to the x-axis so it makes a right angle.

2. Fill in the measures with those of a 45-45-90 triangle

45

A

A

2A

3. Identify the sin cos and tan of the triangle you just created.

sin45

cos45

tan45

2A

A12

22

2A

A12

22

AA 1

TRIGONOMETRY

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x

y

60

Reference Angles



302A

A

3A


sin60

cos60

tan60

32AA

32

2AA

12

3AA 3

Use reference angles to find the sin, cos and tan of 60 degrees.

TRIGONOMETRY

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x

y

60 120

Reference Angles



30 2A

A

3A


sin120

cos120

tan120

32AA

32

2AA

12

3AA 3

Use reference angles to find the sin, cos and tan of 120 degrees.

TRIGONOMETRY

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x

y

45

Reference Angles

45A

A

2A

2sin45 2

2cos45 2

tan45 1

y

45 135

45

A

A 2A

2sin135 2

2cos135 2

tan135 1

TRIGONOMETRY

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Reference Angles

1. sin .32x

Inverse trig functions

2. sin 0.75x

3. cos 1/ 3x

4. sin 0.5x

5. sin 3.6x

1sin (.32) 18.7 161.3or

1sin (.75) 48.6 131.4or

1cos (1/ 3) 70.5 289.5or

1sin (.5) 30 150or

1sin (3.6) ????

Two angles with the same sine add to 180

Two angles with the same cosine add to 360

Since the hypotenuse is the longest side, sine and cosine are always between 1 and -1

Technically, if they are both more than 180, they

add to 540.

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Find the area of this triangle:

3

4

512A b h

1 3 42

6

This formula works for any right triangle, but…

AREA

TRIGONOMETRY

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9

12

17

( )( )( )( )A s s a s b s c Where “S” is the semiperimeter or half the perimeter and a, b, and c are side lengths

HERON’s FORMULA

1(9 12 17)2s 19

(19)(19 9)(19 12)(19 17)A

(19)(10)(7)(2)A

2660A

51.6A

AREA

Half of the perimeter

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8

16

10

( )( )( )( )A s s a s b s c Where “S” is the semiperimeter or half the perimeter and a, b, and c are side lengths

HERON’s FORMULA

1(8 10 16)2s 17

(17)(17 16)(17 8)(17 10)A

(17)(1)(9)(7)A

1071A

32.7A

THIS WORKS IF WE KNOW ALL 3 SIDES, BUT…

AREA

TRIGONOMETRY

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8068

17

AREA

But you don’t know h.

You can figure it out:From the 68, h is the opposite and 8 is the hypotenuse. So use sine!

h

sin68 8h

8sin68h 1 172A h

1 17 8sin682A

63.0A

Put these together and get…

12A b h

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Rather than repeat all this for each problem, we can create a formula

2s

1s

AREA

h

2sin h

s 1

2A b h

112A s h 2 sinh s

1 21 sin2A s s

It is always 2 sides and the sine of the angle between

them.

1 21 sin2A s s

1 sin2A ab C

A

B C

c b

a

Two sides and the angle in between

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AREA

1 sin21 sin21 sin2

Area ab C

ac B

bc A

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Rather than repeat all this for each problem, we can create a formulaAREA

12 2080

1 12 20 sin802A

1 sin2A ab C

118.2

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AREA

30 35

70

1 sin2A ab C

WE CANT DO THIS ONE!

We don’t have 2 sides and the angle between them

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AREA

30

35

701 35 30 sin702A

1 sin2A ab C

493.3

When you’re given the area and two sides, and asked to find the angle between…

The area of a triangle is 60. Two of the legs are 15 and 20. what is the angle in between them?

15

20?

60 = ½ 15 x 20 sin (x)0.4 = sin (x)X = sin-1 (0.4)X = 23.6

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AREA

OR X = 156.4

15

20?

15

20?

1520

?

You can get 2 solutions anytime you are given the area and two sides.

Sometimes you can eliminate one of the solutions if you are given another angle, and two angles in the triangle add to more than 180.

TRIGONOMETRY

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AREA

TRIGONOMETRY

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1 sin21 sin21 sin2

Area ab C

ac B

bc A

A

B Ca

c b

1 sin2Area ab C1 sin2bc A

1 sin2ac B

TRIGONOMETRY

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1 1 1sin sin sin2 2 2ab C ac B bc A

12abc 1

2abc12abc

sin sin sinA B Ca b c

sin sin sina b cA B C

OR

The LAW of SINES

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sin sin sinA B Ca b c

sin sin sina b cA B C

OR

The LAW of SINES

Use the law of sines to find x.

50 72

23 X

sin50 sin7223x

23sin50 sin72x

23sin50sin72 x

18.5

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Find x.

44

30

81X

30sin81 sin44

x

21.1

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8sin84 sin43

x

5.5

10sin32 sin65

x

17.1

8625

sin41 sin86x

38

86

41sin62 sin86

x

46.3

TRIGONOMETRY

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m Aa

m Bb

m Cc

Solve this triangle.

A

B C

41

62

28

41

62

77

7728

28sin41 sin62a

20.8a

20.8

20.8

28sin77 sin62c

30.9c

30.9

30.9

Could we have Used

instead?

20.8sin41

TRIGONOMETRY

Law of Sines

A

B C


4012

16

m Aa

m Bb

m Cc

40

12

16

sin sin4016 12B

sin 0.8570501B 059B

059 or 0121

081 or 019

sin81 sin4012c

18.4c

sin19 sin4012c

6.1c

18.4 or 6.1

2 Solutions

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TRIGONOMETRY

Law of SinesHow can there be 2 solutions?

A 30

10 8

A 30

10 8ANIMATION

2 Solutions

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TRIGONOMETRY

Law of Sines

A 30

10 8

A 30

10 8ANIMATION

A 30

10 8

BOTH of these triangles are solutions

2 Solutions

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TRIGONOMETRY

Law of SinesHow do I know when there are 2 solutions?

A 30

10 8

A 30

10 8ANIMATION

You have to check any time you take the sin -1

A 30

10 8

A 30

10 82 Solutions

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TRIGONOMETRY

Law of Sines

A

B C


20

5517

m Aa

m Bb

m Cc

5517

20

sin sin5520 17A

sin 0.96370829A 074.5A

074.5 or 0105.5

050.5 or 019.5

sin50.5 sin5517b

16b

sin19.5 sin5517b

6.9b 16 or 6.9

2 Solutions

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TRIGONOMETRY

Law of Sines

A

B C


315240

m Aa

m Bb

m Cc

52

40

31

sin52 sin40 31

A

sin 0.610708334A 037.6A

037.6 or 0142.4

090.4 or ???sin52 sin90.4

40 c 50.8c

You don’t have to worry about the second solution here because the angles would add up to more than 180!

50.8Why does this one have only 1 solution when there are 2 angles with the same sine?

Click here or hereA 37

5 3A 37

512

1 Solution

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TRIGONOMETRY

Law of Sines

A

B

C


12218

12

m Aa

m Bb

m Cc

122

18

12

sin122 sin18 12

B

sin .5653653974B 034.4B

034.4 or 0145.6

023.6 or ???

sin122 sin23.618 c

8.5c

8.5

1 Solution

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TRIGONOMETRY

Law of Sines

A

B C


513824

m Aa

m Bb

m Cc

2438

51

sin38 sin24 51

B

sin 1.30828B No Solution

Sine (and Cosine) are NEVER more than 1!

What does a triangle with no solution look like?

A 37

5 1

Click here to find out.

No Solution

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TRIGONOMETRY

Law of Sines

Summary

LAW of SINES: SummaryWhen do we use the law of sines?

Anytime we are solving a triangle and know an angle and the side across from it.We can use it in a right triangle but usually use soh cah toa instead

When might I get no solutions or 2 solutionsAnytime you use sin -1

When using sin -1 we get …0 solutions 1 solution 2 solutionssin -1 (1.2)

The missing side is too short to reach the rest of the triangle

The angles in the second set of solutions add to more than 180.

You get the second set of solutions when you find the first unknown angle.

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i. introduction (from geometry) ii. word problems and alt trig functions

Documents

angle of elevation

right angle

sin cos

ft tall

degree anglexy10we

degree anglexy45now

values of sin

ft high