Download Unit 4: Right Triangles

Unit 4: Right Triangles
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Triangle Inequality
Pythagorean Theorem and its Converse
Trigonometry
Inverse Trigonometry
Solving Right Triangles
Lesson 4.1
• Triangle Inequality
• Converse of the Pythagorean Theorem
• More Classifying Triangles
Triangle Inequality
• The sum of the lengths of any two sides of
a triangle is greater than the length of the
third side.
Practice the Triangle Inequality
•
Can the following lengths represent the sides of
a triangle?
1) 5, 4, 3
Yes, add any two together and they are larger
than the third side.
2) 5, 6, 7
Yes, add any two together and they are larger
than the third side.
3) 5, 5, 10
No, 5+5 is equal to 10, not greater than 10.
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
Pythagorean Theorem
• c2 = a2 + b2
– c is always the hypotenuse
– a and b are the legs in any order
a
c
b
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
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
c
b
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
c
b
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
Triangle Y/N
B) 10.5, 36.5, 37.5
c
longest side
c2 = a2 + b2
862 = 382 + 772
7396 = 1444 + 5929
= 7373
7396 >
obtuse
Triangle Y/N
c2 = a2 + b2
37.52 = 10.52 + 36.52
1406.25 = 110.25 + 1332.25
= 1442.5
1406.25 <
acute
Right Triangle?
Lesson 4.3
Trigonometric Ratios
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.
Sine
• The sine is a ratio of
– 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
θ
a
c
b
sin θ =
side opposite θ
hypotenuse
b
= c
Cosine
• The cosine is a ratio of
– 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
θ
a
c
b
cos θ =
side adjacent θ
hypotenuse
=
a
c
Tangent
• The tangent is a ratio of
– 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
θ
c
b
tan θ =
side opposite θ
side adjacent θ
=
b
a
SOHCAHTOA
S in
o pposite
h ypotenuse
C os
a djacent
h ypotenuse
T an
o pposite
a djacent
• 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
Example 5
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.
• 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.
7
4
x
sin 42o =
42o
x
7
7 (sin 42o) = x
7 (.6691) = x = 4.683
37o
x
cos 37o =
x (cos 37o) = 4
4
x
Get x out
of
denominat
or first by
multiplying
both sides
by x.
4
4
x = cos 37o= .7986 = 5.008
Lesson 4.4
Inverse Trigonometric Ratios:
Solving for missing angles in a right
triangle.
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
Example
• 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
7
4
9
θ
17
θ
4
7
θ = sin-1
θ = sin-1 .5714
sin θ =
θ = 34.8o
4
7
9
17
θ = cos-1 9
17
θ = cos-1 .5294
cos θ =
SOHCAHTOA
θ = 58.0o
Lesson 4.5
• Solving Right Triangles
Solving a Triangle
• To solve a right triangle, you must find
– all 3 sides
– all 3 angles
• or the other 2 angles besides the right angle
• So your final answer when solving a right
triangle will have six parts to the answer!