Trig PowerPoint

8/12/2019 Trig PowerPoint

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!

Opposite

Adjacent

Hypotenuse

Before you learn about trigonometry, we need a common language to talk about it.

Basic trigonometry involves right triangles. Our angle measured degrees,called theta, determines the names of the sides. For example, side is directlyopposite to angle .

is adjacent to side , meaning its directly beside

The hypotenuse is directly across from the right angle

!

A

B

C

AB

! ACB

! ACB

! ACB

AC ! ACB

! BAC


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!

Opposite

Adjacent

Hypotenuse

A

B

C

To the left, you see the image fromthe last slide. What would happenif we changed our viewpoint from

to ?

Clearly, the hypotenuse would stillbe across from the right angle. But,

what would happen to the oppositeand adjacent sides?

! ACB ! ABC

A

B

C

Adjacent

Opposite

Hypotenuse

In the new image to the right, weswitch the location of theta. Whathappens?

As expected, the hypotenuseremains the same.

But, now side is opposite oftheta and side is adjacent totheta.

AC

AB

!


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A ratio expresses relationships between values.

Suppose there are 50 parking spaces in a parking lot,

but 20 are occupied. The ratio of occupied parkingspaces to free parking spaces could be written as 20/50,or , which can be reduced to .

Similarly, we can express ratios between side lengths of

triangles.

20

502

5

Suppose the angle measuring 60 degrees is our concern.

OppositeHypotenuse

=

7 314

AdjacentHypotenuse

=

714


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We have names for these ratios called sine, cosine, andtangent. Below, you will find a list of abbreviations

Therefore, for the triangle to the right, we have:

sin(angle measure) =opposite

hypotenuse

cos(angle measure) =adjacent

hypotenuse

tan(angle measure) =oppositeadjacent

sin(60) =7 3

14 , sin(30) =

7

14

cos(60) =7

14 , cos(30) =

7 3

14

tan(60)=

7 3

7 , tan(30) =

7

7 3


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In the last problem, we knew the triangles side lengths and its angle measures. However, we will often not have all this information.

Youve probably found measurements on triangles given a limited amount of information.Consider the Pythagorean theorem:

Given: 2 side lengths of a triangleFind: The length of the third sideExample:

By the Pythagorean Theorem, the third side the hypotenuse has length 13 units.

With trigonometry, you will have questions that give you 1 side length and 1 angle, whereas this example gives you 2 side lengths. Lets see a trig example in the next slide

5

12

a 2 + b 2 = c 2

5 2 + 12 2 = c 2

25 + 144 = c 2

169 = c 2

169 = c 2

13 = c


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Suppose you must direct your vision l35degrees above the horizontal to see thetop of a tree. How far are you from the

tree?

First, take into account what you know:1) An angle of 35 degrees and

2) the side length opposite to that 35

degree angle is 30 ft.

Second, consider what you want toknow: the length you are from the tree in other words, the length of line CB.Note that line CB is adjacent to angle C.

Now, what trigonometric functionrelates opposite side to adjacent side? If

you review sine, cosine and tangent, youll find that tangent works.

( A ) tan(35) =30

x

(B) x

1* tan(35) =

30 x

* x

1

(C) x tan(35) = 30

(D) x tan(35)tan(35)

=

30tan(35)

(E) x = 30tan(35)

But now what?

See nextslide..


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In the last slide, we found that the distance to the tree was , but what does this mean?

We can calculate this value in our calculator, but we must be

very careful.

Most likely, you have only learned about angles in terms ofdegrees. We talk about a triangles angles adding to 180degrees, or a circle being 360 degrees.

Just like temperature can be measured in Fahrenheit orCelsius, angles can be measured in degrees and radians. You

will learn about radians later, but not in this class. Make sure your calculator is in degree mode and not in radian mode.

Hit the MODE button and then drag your curser over the

word degree, then click Enter youll find. You may findthat the calculator is already in degree mode.

At that point, you can type 30/tan(35) and obtain42.84 .

x =30

tan(35)


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Note that there are multiple ways to find the samemissing side length. For example, consider the twomethods of finding the length y in this righttriangle. I can use the 30 degree angle measure, orthe measure of angle B, which must be 60 degrees.

( A) tan(30) =5 y

(B) y

1 * tan(30)=

5 y *

y

1

(C) ytan(30) = 5

(D) y tan(30)tan(30)

= 5tan(30)

(E) y =5

tan(30)= 8.66

( A) tan(60) = y5

(B)51

*tan(30) = y

5*

51

(C) 5tan(30) = y

(D) 2.89 = y


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5

2.89

30

60 x

In the last slide, you saw how tangent could be usedto find that side length y is 2.89 units. Now, youcould find x by the Pythagorean theorem, or with a

trigonometric equation. Here are several possibletrigonometric equations you could use:

( A ) sin(30) =5 x

(B) x

1*sin(30) = 5

x*

x

1

(C) xsin(30) = 5

(D) x sin(30)sin(30)

=

5sin(30)

(E) x =5

sin(30)= 10

( A ) cos(30) =2.89

x

(B) x1

*cos(30) = 2.89 x

* x1

(C) xcos(30) = 2.89

(D) x cos(30)

cos(30)=

2.89

cos(30)

(E) x =2.89

cos(30)

= 3.34

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