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Chapter 9 Right Triangle Trigonometry
Goals
9.2 The Pythagorean Theorem



Use the Pythagorean Theorem correctly to solve for a missing side of a right triangle, if
given the other two side lengths.
Prove the Pythagorean Theorem.
Identify Pythagorean triples.
9.3 Converse of the Pythagorean Theorem

Classify triangles as acute, right, or obtuse by using the Pythagorean Theorem when
given the three side lengths of a triangle.
9.4 Special Right Triangles


Use special triangle pattern for a 45-45-90 isosceles triangle to quickly find unknown
side lengths.
Use special triangle pattern for a 30-60-90 triangle to quickly find unknown side lengths.
9.5 Trigonometric Ratios





Define sine, cosine, and tangent ratios.
Correctly label the sides of a right triangle as opposite, adjacent, and hypotenuse given
one acute angle.
Set up a trigonometric ratio (sine, cosine, tangent) using two sides of a right triangle and
a given acute angle measure.
Set up a trig equation and solve for a missing side length given one acute angle of a right
triangle, and one side length.
Use the relationship between sine and cosine to compare sides and angles of triangles.
9.6 Solving Right Triangles


Solve for a missing acute angle in a right triangle given two of the side lengths.
Use inverse trig functions to solve the equation for a missing angle measure.
 Use the appropriate trig function (regular or inverse) to solve a right triangle completely.
Chapter 9 Vocabulary
9.2
Pythagorean Theorem
Pythagorean Triple
9.3
converse
9.4
short leg
long leg
hypotenuse
base angles
9.5
trigonometric ratio
sine (sin)
cosine (cos)
tangent (tan)
angle of elevation
angle of depression
9.6
solving a right triangle
arcsine (sin-1)
arccosine (cos-1)
arctangent (tan-1)
Simplifying Square Roots
To simplify a square root, you “take out” anything that is a perfect square.
Example: √63
What is the biggest perfect square that can be divided out of 63?
√63 = √9 ∙ √7 = 3 ∙ √7 = 3√7
Let’s practice a few more.
1. √18
2. √72
3. √112
4.
√132
5. √14  √6
“Three radical seven”
9.2 Notes: The Pythagorean Theorem
Pythagorean Theorem – USED FOR RIGHT TRIANGLES ONLY!!!
If we have a right triangle, then 𝑎2 + 𝑏 2 = 𝑐 2 , is used to find the third side of a triangle if we
are given two known side lengths.
a, b represent the two legs of the triangle, they form the right angle
c represents the hypotenuse of the triangle, across from the right angle (always longest)
A ________________________________ ________________________ is a set of three positive integers,
a,b, and c that satisfy the equation 𝑎2 + 𝑏 2 = 𝑐 2 . For example:
Examples:
1.
Try on your own:
4.
2.
3.
5.
6. Find the area of the triangle.
First, find the height.
Then, find the area of the triangle using base and height.
9.3 The Converse of the Pythagorean Theorem
We know the Pythagorean Theorem, but how can I tell if I have a right triangle?
Converse of Pythagorean Theorem
If 𝑎2 + 𝑏 2 = 𝑐 2 , then we have a right triangle. The converse of the Pythagorean Theorem is
used to determine if a triangle is right, obtuse or acute given 3 side lengths.
Practice:
Is this a right triangle? If not, is it obtuse or acute?
FOLLOW THE HYPOTENUSE!!!! IS IT TOO SMALL? TOO BIG? OR JUST RIGHT?
1.
2.
Practice:
Can the following lengths represent the 3 sides of a triangle? If yes, what kind of triangle
can be formed?
3.
√250, 30, 34
4.
ha ha !
8, 13, 22
9.4 Notes: Special Right Triangles
The 45-45-90 Triangle
Hypotenuse = leg ∙ √𝟐
To find the LEGS – divide by √𝟐
To find the HYPOTENUSE – multiply by √𝟐
1.
2.
GI = ____
HG = ____
3.
4.
5. There is a square window with a diagonal length of 36√2 inches.
What is the area of the window?
The 30-60-90 Triangle
hypotenuse = short leg ∙ 𝟐
Long leg = short leg ∙ √𝟑
ALWAYS FIND THE SHORT LEG FIRST!!!
5.
6.
7.
8.
9.
9.5 Notes: Trigonometric Ratios
I. Introduction
Every right triangle has two _______________ angles. From each of these acute angles, we can
label the three sides of the triangle.
Label the following triangle.
Note: We never label from the _______________ angle. If you change your angle, your labels will
change!
Three basic trigonometric ratios (Memorize!!)
Sine =
Cosine =
Tangent =
SOH CAH TOA
Let’s practice setting up the ratios and simplifying the fractions.
1.
2.
3.
sin R =
sin A =
sin X =
cos R =
tan A =
cos X =
tan R =
cos B =
sin Y =
tan S =
tan B =
cos Y =
II.
Relationships between Sine & Cosine. (DEGREE MODE!!!)
1. sin 40 =
2. cos 36 =
cos 50 =
sin 54 =
The sine & cosine of ___________________________________ angles are always _______________________!
Complementary angles add to __________. Therefore, we can define them in terms of the
other.
𝑚∠𝐴 =
𝑚∠𝐷 =
𝑚∠𝐶 =
𝑚∠𝐹 =
𝑠𝑖𝑛 𝐴 =
cos 𝐷 =
sin 𝐶=
cos 𝐹 =
_________________________________________________________________________________________________
Practice Problems:
1. An equation is shown. sin(𝑎°) = cos(𝑏°)
0 < 𝑎 < 90 and 0 < 𝑏 < 90
Write an expression for b in terms of a.
2.
b = __________________________
III. Solving for missing side lengths using sin-cos-tan
Setting up equations:
 First, always label triangle from your given angle.
o Opposite, adjacent, hypotenuse.
 Then ask yourself:
o “What side was I given?”
o “What side am I trying to find?”
 Identify which ratios to use from these questions!
 Set up correct equation. (Remember to include your angle measure)
 Solve equation using normal inverse operations and answer the question.
Practice.
1. Solve for x
2. Find the height of the pole.
3. Find the hypotenuse
4. Solve for s and r
5. Solve for the missing length, x.
9.6 Notes: Solving Right Triangles
I. Solving for Missing Angles using Sin-Cos-Tan
Inverse Trigonometric Functions

The inverse of sin A is __________________________ or ________________.

The inverse of cos A is _________________________ or ________________.

The inverse of tan A is __________________________ or _______________.
To solve or missing angles using the inverse trigonometry functions:
1. Set up equations the same as before but notice your angle will be a variable.
2. Solve using inverse operations.
Practice finding both angle measures.
a.
b.
II. Solving Right Triangles
When we solve a right triangle, we find all three _________________________________________ and all
three _____________________________________________.
Let’s solve a few. (Round decimals to the nearest tenth)
1.
2.
Solving Right Triangles Chart
I HAVE
I WANT
I WILL USE
2 Sides
3rd Side
Pythagorean Theorem
(9.2)
1 side of 45-45-90 triangle
Other two sides
1 side of 30-60-90 triangle
Other two sides
(9.4)
(9.4)
1 side and 1 angle (other
than 90 degree angle)
2 sides
Another side
Sin-Cos-Tan (9.5)
An angle
Inverse (or arc)
Sin-Cos-Tan (9.6)
Remember this is only for right triangles! If you don’t have a right angle, these methods won’t work!