Right Triangle: c2 = a2 + b2 45-45-90

Acute Triangle: c2 < a2 + b2

Obtuse Triangle: c2 > a2 + b2

30-60-90

Pythagorean Converse

c2 a2 + b2

Special Right Triangles

Angles of

Elevation &

Depression

Sin Θ = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑒𝑢𝑠𝑒 Cos Θ =

𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡

𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑒𝑢𝑠𝑒 Tan Θ =

𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡

Sin A =

Cos A =

Tan A =

Sin B =

Cos B =

Tan B =

Sin C =

Cos C =

Tan C =

To find an angle, use Inverse Trig Functions: sin–1 cos–1 tan–1

Leg Rule

SOH CAH TOA

Altitude Rule

Right Triangles: Formulas and Properties

Graphic Organizer/ Reference (p.1)

Pythagorean Theorem

hypotenusetheisc

legsareb&a

;cba222

Converse of the

Pythagorean Theorem

;cba:obtuse

;cba:right

;cba:acute

222

222

222

Pythagorean Triple

A set of three whole numbers that solve the

Pythagorean Theorem. (3-4-5 is the most common.)

A

B C

c b

a

45-45-90 A

B C n

n2n

30-60-90 A

C B

n2

n

3n30

o

60o

45o

45o

Right Triangles: Formulas and Properties

Graphic Organizer/ Reference (p. 2)

Trigonometry

AB

ACBsin

AB

BCAsin

hypotenuse

oppositesine

A

B C

c b

a

Law of Sines

A

B C

AB

BCBcos

AB

ACAcos

hypotenuse

adjacentcosine

BC

ACBtan

AC

BCAtan

adjacent

oppositeangentt

c

Csin

b

Bsin

a

Asin

c b

a

Law of Cosines

Ccosabbac

Bcosaccab

Acosbccba

2

2

2

222

222

222

A

B C

c b

a

© ("Secondary Math Shop) 2013

SOH – CAH – TOA

S – Sine

O – Opposite

H – Hypotenuse

hyp

oppsin

C – Cosine

A – Adjacent

H – Hypotenuse

T – Tangent

O – Opposite

A – Adjacent

side opp B

side adj A

side opp A

side adj B

hypotenuse

A

C B

***Reminder***

Hypotenuse: the side opposite the right angle

Adjacent: the side “attached” to the acute angle that is NOT the hypotenuse

Opposite: the side that does not form the acute angle

© ("Secondary Math Shop) 2013

We are given the angle, the opposite

and the hypotenuse so that this is a

sine problem.

calculatorinputx

bysidesbothmultiplyx

ratiotrigupsetx

7.4

836sin8

836sin

We are given the angle, the adjacent and

the hypotenuse so that this is a cosine

problem.

calculatorinputx

anglethefindtoinversethetakex

ratiotrigupsetx

2.48

cos)6

4(cos

6

4cos

1

We are given the angle, the opposite

and the adjacent so that this is a

tangent problem.

calculatorinputx

bysidesbothmultiplyx

ratiotrigupsetx

0.6

550tan5

550tan

We are given the angle, the opposite

and the hypotenuse so that this is a

sine problem.

calculatorinputx

bysidesbothdividex

multiplycrossx

ratiotrigupsetx

7.7

70sin70sin

2.7

2.7)70(sin

2.770sin

Similarity in Right Triangles

-Geometric Mean-

Type 1 Type 2 Type 3

x

x

x

h1

h2 h2 h2

h1 h1