similar triangles & trigonometry

TRIGNOMETRY & SIMILAR TRIANGLES

Trigonometry The Tangent Ratio

On a right-angle triangle, the 3 sides each have a name, based on their position.

You will always work from one of the small angles. The side names describe how far the sides are from this angle.

60º adjacent

opposite (If you start working with the second small angle, you have to rename your sides.)

30º adjacent

opposite

"next to" the given angle

"other side" of the triangle from the given angle


In the same way, if we know what the ratio of the two sides are for a right triangle of any angle measure, and we know one side, then we can use a proportion to solve for the other side.

63 opp

h

ratio of adj = 0.466

25º

63(0.466 ) = h

29.4 = h

0.466 = h 63

opposite

adjacent

Your calculator can give you a value for the ratio of these two sides on your triangle, for ANY known angle.

tan 25º = 0.466 tan 2 5 = angle measure

0.4663076


So your process for each of these problems becomes...

Look at your diagram or read the desciption. Determine which side is the opposite side, which side is the adjacent side.

tan 6 2 = 1.8807265 angle measure tan 62º = 1.881

Fill this value in, and solve for x.

1

Write down the tan ratio. 2

Fill in as much of the tan ratio as you can. 3

Use the calculator to figure out the ratio value for the given angle.

4

5

adj tanθ º = opp

62º 12

x

12 tan 62º = x

1.881 = x 12

12(1.881) = x

22.6 = x

Trigonometry The Sine and Cosine Ratios


There are three trig ratios, Sine, Cosine and Tangent.

sinθ = opp cosθ = adj tanθ= opp hyp hyp adj

Which ratio you use depends on which two sides of your triangle you are working with (you will be given one, and asked to find another).

x 30 θº θº θº

40 x

Name the two sides, and decide which ratio you should use to find x.


The Process: Name the 2 sides (given side & unknown side)

Pick the trig ratio.

Substitute the sides

36.9º 40

x 36.9º

x 36.9º

30

1

2

into the ratio. Use the calculator to tan 36.9º

3

find the ratio value for the given angle.

4

Fill this ratio value in,40(0.751)= x and solve for x.

5

40 = adjacent x = opposite

50 = hypotenuse x = adjacent

30 = opposite x = hypotenuse

0.600

x 30

0.600 = 30

sin 36.9º = 30 x

sin 36.9º =

x =

hyp opp

0.800 = 50

50(0.800)= x 40 = x

x

hyp x 50

adj

cos36.9º =

cos36.9º =

30 = x x = 50

x

adj x 40

opp

0.751 = 40

=

tan 36.9º =

Trigonometry Inverse Trig

?


So far we have been solving unknown side lengths. Trig can also be used to solve for unknown angle measures. When we knew the angle measure your calculator could tell you the trig ratio for two side lengths on your triangle.

38º 40

x

x = (40) .0.781 x = 31.24

x

x adj

tanθ = opp

0.781=

tan 38 = 40

40 calculator

step


.

If you know the lengths of two sides, then you can build the ratio and work backwards to find an unknown angle.

θº 40

31.2

tanθ = 0.78 θ = tan 1 (0.78) θ = 38.00

= 31.2

tanθ = opp

40 tanθ

adj


The Process: Name the 2 known sides Pick the trig ratio.

θº 16

11

1

2 Substitute the sides into the ratio and calculate the decimal value.

3

Trig: "the T AN of the ANGLE is the RATIO" Inverse Trig: "the INVE R S E T AN of the R A T IO is the ANGLE" Write the inverse trig statement using the above rule.

Use your calculator to solve for the angle.

4

5

tanθ = 0.688

θ = tan 1 0.688

θ = 34.50

tanθ = 16

tanθ = opp

11 adj

adj opp

This Process works for all three Trig Ratios: S in, Cos, and T an

Trigonometry Word Problems

h θº

d

Trigonometry Word Problems

A couple of terms to know for word problems...

The angle of elevation is the angle at which a viewer looks UP from the horizontal to sight a object.

The angle of depression is the angle at which a viewer looks DOWN from the horizontal to

sight a object.


H

F

D

B

A

C E G I


A

B

C E G I

D

F

H

We can see that angle A is common in ΔABC , ΔADE , ΔAFG and ΔAHI

tan𝐴𝐴 = 𝐵𝐵𝐵𝐵𝐴𝐴𝐵𝐵

= 𝐷𝐷𝐷𝐷𝐴𝐴𝐷𝐷

= 𝐹𝐹𝐹𝐹𝐴𝐴𝐹𝐹

= 𝐻𝐻𝐻𝐻𝐴𝐴𝐻𝐻

The above relationship concludes that ΔABC , ΔADE , ΔAFG and ΔAHI are similar triangles

tan𝐴𝐴 = 𝐵𝐵𝐵𝐵𝐴𝐴𝐵𝐵

= 𝐷𝐷𝐷𝐷𝐴𝐴𝐷𝐷

= 𝐹𝐹𝐹𝐹𝐴𝐴𝐹𝐹

= 𝐻𝐻𝐻𝐻𝐴𝐴𝐻𝐻

Trigonometry Similar Triangles

Similar triangles have the same shape, but not necessarily the same size. Since a triangle's shape is determined by its angles, similar triangles have the same angle measures.

list corresponding vertices in same order

Because all the angles in any triangle add to 180º, if two triangles have a pair of angles that are the same, then all three of their angles are the same, and the triangles are S IMIL AR

A

B

C D

E

F

"is similar to" Δ ABC ~ ΔDEF

In S imilar Triangles,

- corresponding angles are equal, and - the ratios of corresponding sides are equal.

This allows us to set up proportions that will solve for unknown side lengths.

ΔAB C ~ ΔDE F A

B

C D

E

F

a

c

e d b

f 𝑎𝑎𝑑𝑑

= 𝑏𝑏𝑒𝑒 = 𝑐𝑐

𝑓𝑓 ∠𝐴𝐴 = ∠𝐷𝐷 =

∠𝐵𝐵 = ∠𝐸𝐸 =

∠𝐶𝐶 = ∠𝐹𝐹 =

SCALE FACTOR

Scale factor = New Measurement / Old Measurement Old Measurement is sometimes called as “ Base value “ or a reference

Here AB is a line measuring 10m. We take it as the reference or the “Base Value” = 10m

A B

C D

Now there is a new line “CD” which is 5 times the length of AB. How do we find the length of CD ?

SCALE FACTOR

Scale factor = ‘5’ times = 5 So CD = 5 * 10 = 50m

A B

C D

AB is zoomed

out 5times to get CD

SCALE FACTOR If Given CD = 50m and the Scale factor is [ ‘0.2’ times = 1/5 ] We need to find AB ?? So AB = 0.2 * 50 = 10m

A B

C D

CD is zoomed in 5times to

get AB

SCALE FACTOR Since we cant practically show large distances on a piece of paper , we reduce the original length of a measurement by a “ SCALE FACTOR” and represent on a paper. For example , the actual length of the wall in your house maybe 10m , but it is not practical to show and draw the same on a A4 size page which hardly measures 60cm or so. So what we do is we reduce the actual measurement of 10m to a lower value by reducing/zooming in a factor called as “SCALE FACTOR”

SCALE FACTOR

10cm

For example , we draw the side of a lounge with a ruler and measure the same as 10cm. Given Scale factor is 1:100 which means the actual length of the lounge in your house is 100* 10cm = 1000cm = 10m

Typical Problems of Similar Triangles

similar triangles & trigonometry

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