Trigonometry Name - Amazon S3 · 2019. 8. 11. · Trigonometry Name: N o t es Date: Barry is a merchant sailor who has to plan a trip to three different ports. He can only carry enough

Trigonometry Name:

Notes Date:

Barry is a merchant sailor who has to plan a trip to three different

ports. He can only carry enough supplies to go to two ports, before

he must restock at the main port. Barry needs to make a route that

allows him to make deliveries to all three ports and balances the

amount of time the trip will take with the distance he has to

travel. He will need to use trigonometry.

Write the definition of the term and include an image or example that represents it.

Term Definition Example

Sine

Cosine

Tangent

Pythagorean

Theorem

Pythagorean

Triple

Law of Sines

Law of

Cosines

Pythagorean Theorem The Conceptualizer!

In right triangles, you’ve likely had to solve

for a third side when you have the other

two.

To do this you use the Pythagorean

Theorem.

In a right triangle, the triangle with sides

(or legs) a and b and hypotenuse c has the

sides related by:

a2 + b2 = c2

Pythagorean Theorem Notes

A right triangle has legs of 3 and 2. What is

the hypotenuse?

Pythagorean Triples Extra! Extra!

In some of these examples, all the side

lengths are integers. These sets of three

integer values are known as Pythagorean

Triples.

The simplest (and most-used), with the

smallest values, is the 3:4:5 triple.

Other triples with small values are 5:12:13

and 8:15:17.

© Clark Creative Education

Angles in Right Triangles The Conceptualizer!

All triangles have three sides and three

angles. Each side is opposite an angle; each

angle is opposite a side.

We use particular language when discussing

right triangles.

A right triangle has one right angle, of 90°.

The other two angles are both acute angles.

The side opposite that right angle is the

hypotenuse. It is the longest side in the

right triangle.

The other two sides are the legs of the right

triangle.

An acute angle is formed at the vertex of

the triangle that is the intersection of the

hypotenuse and one of these sides. For the

angle created, the non-hypotenuse side is

the adjacent side. (The other side is the

opposite side.)

Right Triangle Trigonometry The Conceptualizer!

Suppose two right triangles are similar. This

means:

(1) Their corresponding sides are in

proportion.

(2) Their corresponding angles are

congruent.

If you double the size of the triangle -- the

angle measures don’t double. They remain

the same.


Trigy Mysteries

In trig problems, you are almost always given two pieces of information and asked to find a

specific third piece of info. The two pieces you are given and the piece of interest decide

which method you must use.

In the previous example: you are given two sides and are asked to find the hypotenuse.

These three pieces together are what make up the Pythagorean Theorem -- so that’s what

you use.

The mnemonic SOH CAH TOA is used when we are given an angle or are in search of one.

ines = oppositehypotenuse osinec = adjacent

hypotenuse angentt = oppositeadjacent

Sine The Conceptualizer!

In cases where you either are given or are

interested in an angle and its opposite side

and the hypotenuse -- we use sine.

ines = oppositehypotenuse

That ratio is called the sine of the angle. It

is abbreviated sin.

in(θ)s = oppositehypotenuse

Sine Notes


Cosine The Conceptualizer!


interested in an angle and its adjacent side

and the hypotenuse -- we use cosine.

osinec = adjacenthypotenuse

That ratio is called the cosine of the angle.

It is abbreviated cos.

os(θ)c = adjacenthypotenuse

Cosine Notes

Tangent The Conceptualizer!


interested in an angle and its adjacent and

opposite sides -- we use tangent.

angentt = oppositeadjacent

That ratio is called the tangent of the

angle. It is abbreviated tan.

an(θ)t = oppositeadjacent


Stretch!

An important point. Imagine that the opposite side of our

triangle is slightly increased, from 3 to 3.1 units.

The opposite angle will grow, too:

.an(57.2°) .55 t = 23.1 = 1

Tangent Notes

Sketch a triangle problem to illustrate when you need to use each situation.

Pythagorean

Theorem Sine Cosine Tangent


Inverse Trigonometric Operations The Conceptualizer!

So what if we are missing the angle? We will

have to do the trig function in reverse.

The calculator just uses its built-in table in

reverse.

On your calculator, you will look for the

inverse trig function buttons. For example,

with tangent it will either read arctan or

.tan−1

The calculator gives you back the angle.

Similarly, use the arcsin to find an angle

when you know the opposite side and

hypotenuse, or arccos when you know the

adjacent side and hypotenuse.

Inverse Trig Functions Notes


Law of Cosines The Conceptualizer!

Can you use trigonometric functions in

triangles that are not right triangles?

Yes, you can, and you would be using the

Law of Cosines. The general form of the

Law of Cosines is:

ab os Cc2 = a2 + b2 − 2 · c

We use this when the

two sides known have

length a and b. The

angle included between

them is C; the side

opposite has length c.

#SASreturns

When the angle is 90°, the term with cosine

drops out, because the cosine of 90° is 0.

You are left with the Pythagorean Theorem.

Law of Cosines Notes

From a point, Sandy measures one line 80 meters to one side of a pond, and another line 92 meters

to the other side of the pond. The two lines make an angle of 25°. How far is it across the pond?


Law of Cosines: SSS The Conceptualizer!

Can you solve a triangle when you have the

three side values, but no angles to work

with?

Yes, use the Law of Cosines, but solve for

the angle. You will just use arccos to find

the angle.

Law of Sines The Conceptualizer!

Can you use the Law of Cosines to find x

here?

Try as you might, you can’t set it up so that

you know either a, b, and angle C, or all

three sides. #ASAreturns

When you have two angles and a side, you

can use the Law of Sines.

The Law of Sines says that the proportion of

each side to the sine of its opposite angle is

the same:

asin(a) = b

sin(b) = csin(c)


Law of Sines Notes

Find the value of x here:

Law of Sines: SSA [Ambiguous] Case The Conceptualizer!

You might know the congruence theorem

HL, “hypotenuse-leg”. Two right triangles

are congruent if they have a matching

hypotenuse and leg.

This is a special case of the side-side-angle

case, SSA.

But SSA is otherwise problematic.

Both of these triangles have the same SSA

specifications, but they are not congruent:

Each has a side of 2.24, a side of 2.83, and

an angle of 45°.


Law of Sines: SSA [Ambiguous] Case The Conceptualizer!

You get this ambiguous case when you have

angle A, opposite side a, and side b,

between them, with and b > a in(b) s

between 0 and 1.

Solving a Triangle Extra! Extra!

To “solve” a triangle, you find all three

sides and all three angles.

Typically, you would use a combination of

Law of Sines and Law of Cosines, more than

once, depending on what starting

information you have.

If I have _______, what can I use to solve the triangle?

SSS ASA, AAS SAS Right Triangle with a

Side and Angle


A ladder is placed against a house so that its base makes an angle of 75°with the ground. If the

ladder touches the house 7.5 feet up, how far away from the house is the base?

Susan can walk up the stairs to climb 4 feet high, or use a ramp 25 feet long. What angle does the

ramp make with the ground?

Two rescue helicopters are 2000 meters from each other when a call comes in. The angle from the

emergency site to Team Alpha and then to Team Bravo is 42°; from the emergency site to Team

Bravo and then to Team Alpha is 25°. What is the distance, in meters, that the closer helicopter

travels to reach the site?


Trigonometry Name - Amazon S3 · 2019. 8. 11. · Trigonometry Name: N o t es Date: Barry is a merchant sailor who has to plan a trip to three different ports. He can only carry enough

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