11-12-13 Pythagorean Theorem, Special angles, and Trig Triangles Right Triangles Test Review.

11-12-1311-12-13 Pythagorean Theorem, Special angles, and Trig Triangles

Pythagorean Theorem, Special angles, and Trig Triangles

Right TrianglesRight TrianglesTest ReviewTest Review

Pythagorean TheoremPythagorean Theorem

TerminologyTerminology

• We call the two sides that touch the right angle the LEGS. They are the a and b.

• The side opposite the right angle is the hypotenuse. It is the c.

• Label each part.

TerminologyTerminology

• The hypotenuse is always the longest side but shorter than the two legs put together.

• Here is the 60 Second Review

Steps to SolveSteps to Solve

• If you know the 2 legs – square the numbers – add- take square root.

Steps to SolveSteps to Solve

• If you know the hypotenuse and one leg – square the numbers - subtract – take the square root.

Special Right AnglesSpecial Right Angles

45 – 45- 90

Special Right AnglesSpecial Right Angles

If we represent the legs of an isosceles right triangle by 1, we can

use the Pythagorean Theorem to establish pattern relationships

between the lengths of the legs and the hypotenuse.  These relationships will be stated as "short cut formulas" that will allow us to quickly arrive at

answers regarding side lengths without applying trigonometric

functions, or other means.There are two pattern formulas that

apply ONLYto the 45º-45º-90º triangle.

Special Right AnglesSpecial Right Angles

Special Right AnglesSpecial Right Angles

Pythagorean Theorem, Special angles, and Trig Triangles

Pythagorean Theorem, Special angles, and Trig Triangles

What if I forget the pattern formulas?What should I do?

Let's look at 3 solutions to this problem where you are asked to find x:

Pattern Formula solutionPythagorean Theorem

solutionTrigonometric solution

We are looking for the hypotenuse so we will use the pattern formula that will give the answer

for the hypotenuse:

Substituting the leg = 7, we arrive at the answer:

A nice feature of the pattern formulas is that the answer is already in

reduced form.

Since a 45º-45º-90º, also called an isosceles right

triangle, has two legs equal, we know that the

other leg also has a length of 7 units.

c2 = a2 + b2

x2 = 72 +72

x2 = 49 + 49x2 = 98

Use either 45º angle as the reference angle.  One

possible solution is shown below:

 rounded rounded

Special Right AnglesSpecial Right Angles

Special Right AnglesSpecial Right Angles

If you draw an altitude in an equilateral triangle, you will form two congruent 30º- 60º- 90º triangles.  Starting with the sides of the equilateral triangle to be 2, the Pythagorean Theorem will allow us to establish pattern relationships between the

sides of a 30º- 60º- 90º triangle.  These relationships will be stated here as "short cut formulas" that will allow us to quickly

arrive at answers regarding side lengths without applying trigonometric functions, or other means.

There are three pattern relationships that we can establish thatapply ONLY to a 30º-60º-90º triangle.

Special Right AnglesSpecial Right Angles

Labeling:H = hypotenuseLL = long leg (across from 60º)SL = short leg (across from 30º)

Again 1st step is to LABEL what you have!!

Short Cut Pattern FormulasShort Cut Pattern Formulas

short leg:

You must remember that these formula patterns

can be used ONLY in a 30º-60º-90º triangle.

hypotenuse:H = 2SL

Long leg:

Easy PracticeEasy Practice

x is the short leg

Answer

y is the long leg

LL = SL  

Answer

Harder PracticeHarder Practice

6 is the short leg andx is the hypotenuse

(start with what you have given)

H = 2 (SL)

x = 2 (6) 

Answer

y is the long legLL = SL

  y = 6

Answer

Hardest Practice ( bonus?)Hardest Practice ( bonus?)

8 is the long leg and x is the hypotenuse(start with what you

have given)

  Answer

x =16/3

y is the short leg

Answer

Right Δ TrigonometryRight Δ Trigonometry

Label Hypotenuse 1st. Then label side opposite of angle. The side left is your adjacent side. Label Hypotenuse 1st. Then label side opposite of angle. The side left is your adjacent side.

Right Δ TrigonometryRight Δ TrigonometryMUST LABEL SIDES FROM

ANGLE POINT OF VIEW

MUST LABEL SIDES FROM

ANGLE POINT OF VIEW

Right Δ TrigonometryRight Δ Trigonometry

Right Δ TrigonometryRight Δ Trigonometry

With both the TI 83 and 84 you need to make sure you are in degrees.

With yellow TI 84, you need a ^^ after the angles to tell the calculator you are in degrees. But you do not need anything to solve for the inverse operations. Zoom equation takes care of it.

Right Δ TrigonometryRight Δ Trigonometry

There are actually 6 functions with trignonmetry. However, the cosecant, secant, and cotangent are just the reciprocal of the sine, cosine, and tangent.

When we do not know the degree of the angle, we can calculate the fraction ratio, then find the inverse of the ratio with the 2nd button on the calculator. This isn’t needed in zoom 400 and above. But in 300, divide the fraction, push 2nd then your 9 sine, cosine, or tangent button, then 2nd and (-) button for the answer of the fraction above.

Angle of ElevationAngle of Elevation

The angle of elevation is always measured from the ground up.  Think of it like an elevator that only goes up.  It is always INSIDE the triangle.In the diagram at the left, x marks the angle of elevation of the top of the tree as seen from a point on the ground.You can think of the angle of elevation in relation to the movement of your eyes.  You are looking straight ahead and you must raise (elevate) your eyes to see the top of the tree.

Angle of DepressionAngle of Depression

The angle of depression is always OUTSIDE the triangle.  It is never inside the triangle.In the diagram at the left, x marks the angle of depression of a boat at sea from the top of a

lighthouse.You can think of the angle of depression in relation to the movement of your eyes.  You are

standing at the top of the lighthouse and you are looking straight ahead.  You must lower (depress) your eyes to see the boat in the water.

As seen in the diagram above of angle of depression, the dark black horizontal line is parallel to side CA of triangle ABC.  This forms what are called alternate interior angles which are equal in measure (so, x also equals the measure of  <BAC).Simply stated, this means that:

the angle of elevation = the angle of depression. 

Let’s PracticeLet’s Practice

Let’s PracticeLet’s Practice

Let’s PracticeLet’s Practice

Let’s PracticeLet’s Practice

Things to RememberThings to Remember