Right Triangle Trigonometry

Chapter 4: Trigonometric FunctionsSection 4.3 (pg. 277-287)

Let’s review some stuff…• If we want to convert degrees to radians we multiply by…

π180

• If we want to convert radians to degrees we multiply by…

180π

A degree is an angle measure, a radian is an arc measure

Let’s review some stuff…What is the Pythagorean Theorem?

• In a right triangle, the length of one leg squared plus the length of the second leg squared equals the squared length of the hypotenuse.

So what does that mean in English?

a2 + b2 = c2

Let’s review some stuff…- A right triangle is a triangle with a right angle. Let’s look at one of the acute angles of the triangle, which we will call theta = θ

θ

- The side opposite the right angle is called the hypotenuse

- The side opposite θ is called the opposite side (makes sense right?)

- The side next to θ is called the adjacent side

Hypotenuse

Adjacent

OppositeUsing the lengths of these 3 sides, you can form six ratios that define the six trigonometric functions of the acute angle θ

Do you remember the 6 trig functions?

sin(x)

cos(x)

tan(x)

cosecant(x)

secant(x)

cotangent(x)

Right Triangle Definitions of Trig Functions

- If θ is the acute angle of a right triangle, then we can find the values of sin(θ), cos(θ) and tan(θ) using the side lengths of the triangle like so:

sin(θ) =

cos(θ) =

tan(θ) =

opposite

opposite

hypotenuse

hypotenuse

adjacent

adjacent

Let’s do an example…

θ

Find:

sin(θ) = =

cos(θ) = =

tan(θ) = =

oppositehypotenuse

oppositeadjacent

adjacenthypotenuse

45

35

43

This is where the mnemonicSOH CAH TOA comes from …

S O H C A H T O AINE

PPOSITE

PPOSITE

YPOTENUSE

YPOTENUSE

OSINE

ANGENT

DJACENT

DJACENT

Right Triangle Definitions of Trig Functions

- If θ is the acute angle of a right triangle, then we can also find the values of cosecant(θ), secant(θ) and cotangent(θ) using the side lengths of the triangle like so:

cosecant(θ) =

secant(θ) =

cotangent(θ) =

hypotenuse

adjacent

opposite

adjacent

opposite

hypotenuse

Let’s do another example…

θ

Find:

csc(θ) = =

sec(θ) = =

cot(θ) = =

hypotenuseopposite

adjacentopposite

hypotenuseadjacent

54

53

34

Right Triangle Definitions of the 6 Trig Functions

Take a few minutes and make some conjectures with your

table group about the relationships between the 6

trig functions

Write down at least one conjecture

FUNdamental Trig Identities

Evaluating Trig Functions- In the earlier example, you were given the side lengths of a triangle and asked to find sin(θ), cos(θ), and tan(θ). Sometimes you will be asked to find sin, cos, tan for a given acute angle, like θ = 45⁰

- How can you find the value of sin(45⁰)?• We could use a calculator: sin(45⁰) = .7071067812

• But that’s just an approximation, what if we wanted the exact value, not the decimal approximation?

We can construct a right triangle with 45⁰ as one of its acute angles

Evaluating Trig FunctionsFind the exact value of sin(45⁰)• We will start by drawing a triangle with 45⁰ as one of its acute angles. The length of the adjacent side will be 1 unit.

1

45⁰

• Now we can use a little bit of Geometry to help us out. Since two angles of the triangle are 90⁰ and 45⁰, the other angle has to be …45⁰

45⁰

• Since the base angles are congruent, the legs must be congruent also, so the other side length is …

1

1

• Finally, the Pythagorean Theorem tells us that the hypotenuse is …

√2

√2

Evaluating Trig FunctionsFind the exact value of sin(45⁰)

1

45⁰

45⁰

1

√2

• Now we are almost done! We can use SOH CAH TOA to find sin(45⁰)

sin(45⁰) = =

opphyp

1√2

Now find cos(45⁰) and tan(45⁰) and check your answer with a calculator

Use your calculator to make sure that this is the answer we got earlier

1

45⁰

45⁰

1

√2

This triangle has a special name …

It is called a 45-45-90 right triangle, and we use it to find sin(45⁰), cos(45⁰) and tan(45⁰)

1

60⁰

30⁰

2

This triangle also has a special nameIt is called a 30-60-90 right triangle, and we can use it to find trig values involving 30⁰ and 60⁰

√3

Find:

tan(60⁰)

cos(30⁰)

sin(30⁰)

cos(60⁰)

Now check your answers with a calculator

Sines, Cosines, and Tangents of Special Angles

Real World Applications…• Pioneer’s Clock Tower is 40 ft high, and you are standing 40 ft away staring at the top of the tower. At what angle is your head looking upward?o Let’s draw a picture. We know you are 40 ft away and the tower is

40 ft high. We will call the angle your head is looking up θ.

θ

40

40

o We have the opposite and the adjacent sides of the right triangle, so we can say:

tan(θ) = 4040 = 1

And we know from our 45-45-90 triangle that θ must be … 45⁰

Real World Applications…

Brendan Gibbons kicks the football at an angle of 60⁰ when he is 45 yards away from the field goal post. How far did the football travel through the air?

60⁰

45 yards

x = distance ball traveled

o Since this is a right triangle, we can set up a trig equation. Should we use sin(60⁰), cos(60⁰), or tan(60⁰)?

cos(60⁰) = adjhyp

45 x

=o And now since cos(60⁰) = ½ we can say…

1 45 2 x

=

o And we solve for x …