1. Unit 4: Right Triangles Triangle Inequality
Pythagorean Theorem and its Converse
Trigonometry
Inverse Trigonometry
Solving Right Triangles

2. Lesson 4.1 Triangle Inequality Converse of the Pythagorean Theorem
More Classifying Triangles

3. Triangle Inequality
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

4. Practice the Triangle Inequality
Can the following lengths represent the sides of a triangle?
5, 4, 3
Yes, add any two together and they are larger than the third side.
2) 5, 6, 7
5, 5, 10
No, 5+5 is equal to 10, not greater than 10.

5. Special Parts in a Right Triangle
Right triangles have special names that go with it parts.
For instance:
The two sides that form the right angle are called the legs of the right triangle.
The side opposite the right angle is called the hypotenuse.
The hypotenuse is always the longest side of a right triangle.
hypotenuse
legs

6. Pythagorean Theorem c a b c2 = a2 + b2 c is always the hypotenuse
a and b are the legs in any order c a b

7. Converse of the Pythagorean Theorem
If c2 = a2 + b2 is true, then the triangle in question is a right triangle.
You need to verify the three sides of the triangle given will make the Pythagorean Theorem true when plugged in.
Remember the largest number given is always the hypotenuse
Which is c in the Pythagorean Theorem

8. Acute Triangles from Pythagorean Theorem
If c2 < a2 + b2, then the triangle is an acute triangle.
So when you check if it is a right triangle and the answer for c2 is smaller than the answer for a2 + b2, then the triangle must be acute
It essentially means the hypotenuse shrunk a little!
And the only way to make it shrink is to make the right angle shrink as well! a b c

9. Obtuse Triangles from Pythagorean Theorem
If c2 > a2 + b2, then the triangle is an obtuse triangle.
So when you check if it is a right triangle and the answer for c2 is larger than the answer for a2 + b2, then the triangle must be obtuse
It essentially means the hypotenuse grew a little!
And the only way to make it grow is to make the right angle grow as well! a b c

10. Practice
Determine if the following sides create triangle. If they do determine if it is a right, obtuse, or an acute triangle.
A) 38, 77, 86
B) 10.5, 36.5, 37.5 c
longest side
Triangle Y/N
Triangle Y/N
c2 = a2 + b2
c2 = a2 + b2
862 =
37.52 =
7396 = =
7396 = 7373
7396 > 7373
< =
obtuse
acute

11. Right Triangle?

12. Lesson 4.3
Trigonometric Ratios

13. Trigonometric Ratios
A trigonometric ratio is a ratio of the lengths of any two sides in a right triangle.
You must know:
one angle in the triangle other than the right angle
one side (any side) of the triangle.
These help find any other side of the triangle.

14. Sine The sine is a ratio of c a b side opposite the known angle, and…
the hypotenuse
Abbreviated
sin
This is used to find one of those sides.
Use your known angle as a reference point a b c θ
side opposite θ b
sin θ = = c
hypotenuse

15. Cosine The cosine is a ratio of c a b
side adjacent the known angle, and…
the hypotenuse
Abbreviated
cos
This is used to find one of those sides.
Use your known angle as a reference point a b c θ
side adjacent θ a
cos θ = = c
hypotenuse

16. Tangent The tangent is a ratio of c a b
side opposite the known angle, and…
side adjacent the known angle
Abbreviated
tan
This is used to find one of those sides.
Use your known angle as a reference point a b c θ
side opposite θ b
tan θ = =
side adjacent θ a

17. SOHCAHTOA S o h C a T in
This is a handy way of remembering which ratio involves which components.
Remember to start at the known angle as the reference point.
Also, each combination is a ratio
So the sin is the opposite side divided by the hypotenuse
pposite
ypotenuse os
djacent
ypotenuse an
pposite
djacent

18. If you do not have a calculator with trig buttons, then turn to p845 in book for a table of all trig ratios up to 90o.
Example 5
First determine which trig function you want to use by identifying the known parts and the variable side.
Use that function on your calculator to find the decimal equivalent for the angle.
Set that number equal to the ratio of side lengths and solve for the variable side using algebra.
37o
42o 4 x x 7
Get x out of denominator first by multiplying both sides by x. x 4
sin 42o =
cos 37o = 7 x
7 (sin 42o) = x
x (cos 37o) = 4 4 = 4
.7986
7 (.6691) = x
= 4.683
x =
= 5.008
cos 37o

19. Lesson 4.4
Inverse Trigonometric Ratios: Solving for missing angles in a right triangle.

20. Finding Angle Measures
Inverse Trig Ratios
Inverse trig ratios are used to find the measure of the angles of a triangle.
The catch is…you must know two side lengths.
Those sides determine which ratio to used based on the same ratios we had from before.
Finding Side Lengths
Finding Angle Measures
sin
sin-1
cos
cos-1
tan
tan-1
SOHCAHTOA

21. Example 8
You still base your ratio on what sides are you working with compared to the angle you want to find.
Only now, your variable is θ.
So once you find your ratio, you will then use the inverse function of your ratio from your calculator θ 4 7 θ 9 17 4 7
sin θ = 9 17
cos θ = 4 7
θ = sin-1 9 17
θ = cos-1
θ = sin
θ = cos
SOHCAHTOA
θ = 34.8o
θ = 58.0o