Trigonometry Power Point

8/8/2019 Trigonometry Power Point

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Basic Trigonometry

Sections of this are from “Trigonometry in aNutshell"

© 2001 The Math Forum @ Drexelremainder by Gary Greer



When you have a righttriangle there are 5 thingsyou can know about it.. the lengths of the sides (A, B, and C)

the measures of the acute angles (a and b)

(The third angle is always 90 degrees)

AC

B

a

b



If you know two of the sides, youcan use the Pythagorean

theorem to find the other side

22

22

22

B AC

AC B

BC A

+=

−=

−=

A = 3C

B = 4

a

b

525

43

4,3

22

22

==

+=

+=

==

C

C B AC

B Aif



And if you know either angle, aor b, you can subtract it from 90

to get the other one: a + b = 90 This works because there are 180º in a

triangle and we are already using up 90º

For example: if a = 30º

b = 90º – 30ºb = 60º

AC

B

a

b



But what if you want to knowthe angles? Well, here is the central insight of

trigonometry:

If you multiply all the sides of a right triangle

by the same number (k), you get a trianglethat is a different size, but which has the

same angles:

k(A)

k(C)

k(B)

a

bA

C

B

ab



How does that help us?

Take a triangle where angle b is 60º and

angle a is 30º

If side B is 1unit long, then side C must be 2

units long, so that we know that for a triangleof this shape the ratio of side B to C is 1:2

There are ratios for every

shape of triangle!

A = 1

C = 2

B

30º

60 º



But there are three pairs of sides possible! Yes, so there are three sets of ratios for any

triangle They are mysteriously named:

sin…short for sine

cos…short for cosine

tan…short or tangent and the ratios are already calculated, you just

need to use them



So what are the formulas?

hyp

opp=θ sin

hyp

adj=θ cos SOH

adjopp=θ tan

CAHTOA

Sin is Opposite over HypotenuseCos is Adjacent over HypotenuseTan is Opposite over Adjacent



Some terminology:

Before we can use the ratios we need to get

a few terms straight

The hypotenuse (hyp) is the longest side of

the triangle – it never changes

The opposite (opp) is the side directly across

from the angle you are considering

The adjacent (adj) is the side right beside theangle you are considering



A picture always helps…

looking at the triangle in terms of angle b

AC

B

b

adj

hyp

opp

b C is always the hypotenuse

A is the adjacent

(near the angle)

B is the opposite (across from

the angle)

LongestNear

Across



But if we switch angles…

looking at the triangle in terms of angle a

AC

B

a

opp

hyp

adj

a

C is always the hypotenuse

A is the opposite (across from

the angle)

B is the adjacent (near the

angle)

LongestAcross

Near



Lets try an example

Suppose we want tofind angle a

what is side A?

the opposite what is side B? the adjacent

with opposite andadjacent we usethe…

tan formula

adj

opp=θ tan

A = 3C

B = 4

a

b



Lets solve it

adj

opp=θ tan

A = 3C

B = 4

a

b

75.04

3tan ==a

scalculator our check

36.87ºa =



Where did the numbers forthe ratio come from? Each shape of triangle has three ratios

These ratios are stored your scientific

calculator

In the last question, tanθ = 0.75

On your calculator try 2nd, Tan 0.75 = 36.87 °



Another tangent example…

we want to find angle b

B is the opposite

A is the adjacent

so we use tan

adj

opp=θ tan

A = 3C

B = 4

a

b

°=

=

=

13.53

33.1tan3

4tan

b

b

b



Calculating a side if you knowthe angle you know a side (adj) and an angle (25°)

we want to know the opposite side

adj

opp=θ tan

AC

B = 6

25°

b

80.2

647.0

625tan

625tan

=

×=

×°=

=°

A

A

A

A



Another tangent example

If you know a side and an angle, you can

find the other side.

adj

opp=θ tan

CA = 6

25°

b

B87.12

47.0

6

25tan

6

6

25tan

=

=

°=

=°

B

B

B

B



An application

65°

10m

You look up at an angle of 65° at the top of a treethat is 10m away

the distance to the tree is the adjacent side

the height of the tree is the opposite side

4.21

14.210

65tan10

1065tan

=

×=

°×=

=°

opp

oppopp

opp



Why do we need the sin &cos? We use sin and cos when we need to work

with the hypotenuse

if you noticed, the tan formula does not have

the hypotenuse in it. so we need different formulas to do this work

sin and cos are the ones!

C = 10

A

25°

b

B



Lets do sin first

we want to find angle a

since we have opp and hyp we

use sin

hyp

opp

=θ sin

C = 10

a

b

B

A = 5

°=

=

=

30

5.0sin

10

5sin

a

a

a



And one more sin example

find the length of side A

We have the angle and

the hyp, and we need

the opp

hyp

opp=θ sin

C = 20

25°

b

B

A

45.8

2042.0

2025sin

2025sin

=

×=

×°=

=°

A

A

A

A



And finally cos

We use cos when we need to work with thehyp and adj

so lets find angle bhyp

adj=θ cos

C = 10

a

b

B

A = 4

°=

=

=

42.66

4.0cos

10

4cos

b

b

b

°=

°°=

23.58a

66.42-90a



Here is an example

Spike wants to ride down a steelbeam

The beam is 5m long and is

leaning against a tree at an angleof 65° to the ground

His friends want to find out how

high up in the air he is when he

starts so they can put add it to the

doctors report at the hospital

How high up is he?



How do we know whichformula to use??? Well, what are we working with?

We have an angle

We have hyp

We need opp

With these things we will use

the sin formula

C = 5

65°

B



So lets calculate

so Spike will have fallen 4.53m

C = 5

65°

B

53.4

591.0

565sin

565sin

65sin

=

×=

×°=

=°

=°

opp

opp

opp

opp

hypopp



One last example…

Lucretia drops her walkman

off the Leaning Tower of

Pisa when she visits Italy

It falls to the ground 2meters from the base of the

tower

If the tower is at an angle of 88° to the ground, how far

did it fall?



First draw a triangle

What parts do we have?

We have an angle

We have the Adjacent

We need the opposite

Since we are working with

the adj and opp, we will use

the tan formula

2m

88°

B



So lets calculate

Lucretia’s walkman fell 57.27m

2m

88°

B

27.57

264.28

288tan

288tan

88tan

=

×=

×°=

=°

=°

opp

opp

opp

opp

adj

opp



What are the steps for doingone of these questions?1. Make a diagram if needed

2. Determine which angle you are working with

3. Label the sides you are working with

4. Decide which formula fits the sides

5. Substitute the values into the formula

6. Solve the equation for the unknown value

7. Does the answer make sense?



Two Triangle Problems

Although there are two triangles, you only

need to solve one at a time

The big thing is to analyze the system to

understand what you are being given Consider the following problem:

You are standing on the roof of one building

looking at another building, and need to findthe height of both buildings.



Draw a diagram

You can measure the

angle 40° down to

the base of other

building and up 60° to the top as well.

You know the

distance between the

two buildings is 45m

60°

40°

45m



Break the problem into twotriangles. The first triangle:

The second triangle

note that they share a

side 45m long a and b are heights!

60°

45m

40°

b

a



The First Triangle

We are dealing with an angle, the opposite

and the adjacent

this gives us Tan

60°

45m

a

77.94ma

451.73a

4560tan

4560tan

=

×=

×°=

=°

a

a



The second triangle

We are dealing with an angle, the opposite and

the adjacent

this gives us Tan

45m

40°

b

37.76m b

450.84 b

4540tan

4540tan

=

×=

×°=

=°

b

b



What does it mean?

Look at the diagram now:

the short building is

37.76m tall

the tall building is 77.94m

plus 37.76m tall, which

equals 115.70m tall60°

40°

45m

77.94m

37.76m



That’s all for Trigonometry

Sections of this are from “Trigonometry in aNutshell"

© 2001 The Math Forum @ Drexelhttp://www.mathquest.com/library/drmath/drmath.high.html

remainder by Gary Greer

http://www.mathquest.com/library/drmath/drmath.high.html




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Trigonometry Power Point