45⁰

45 – 45 – 90 Triangle:

The hypotenuse is 2 times the hypotenuse.

60 ⁰

30 – 60 – 90 Triangle:

i) The hypotenuse is twice the shorter leg.

ii) The longer leg is 3 times the shorter leg.

TRIANGLE TRIGONOMETRYI) Right Triangle Trigonometry

C

B

A

Hypotenuse

Side opposite B

Side adjacent to B

Pythagorean TheoremIn any right triangle,

c2 = a2 + b2

A C

B

c

b

a

ACb

B

ca

length of side oposite angle A sin A =

length of hypotenuse

length of side adjacent to angle Acos

length of hypotenuse

length of side oppo

TRIGONOMETRIC RATIO

site angle Atan

length of side adja

S

a

c

bA

c

A

cent to angle A

a

b

Example 1: Find the three trigonometric ratios for angle A in the triangle.

144 m

156 m

60.0 mC A

B

length of side oposite angle Asin A =

length of hypotenuse

length of side adjacent to angle Acos

length of hypotenuse

length of side opposite angle Atan

length of side adjacent to angle A

a

c

bA

c

aA

b

144 m

156 0. 1

m923

60.0 m

1560.

m6

384

144 m =

60 m2.400

We can find the measures of angles A, B, and C byusing a trig table or the second function on a calculator.

1 144 mA sin ( )

15667

m.4

We know: A + B + C = 1800, and C = 900

therefore, A + B = 900

B = 900 – 67.40 = 22.60

Mnemonic device for remembering sine, cosine, and tangents ratios:

SOHCAHTOA

ppositeine

ypotenuse

OS

H

djacentosine =

ypotenuse

AC

H

ppositeangent =

djacent

OT

A

SOLVING RIGHT TRIANGLES

To solve a triangle means to find the measures of the various parts of a triangle that are not given.

Examples: Draw, label, then solve the right triangle described below. C is the right angle.

1)B = 36.70, b = 19.2m

A = 53.30, c = 32.1m, and a = 25.8 m

BC

A

c

a

b

2) Find AC in right triangle ABC. a = 225m, and A = 10.10 . C is the right angle. (Don’t use the Pythagorean Theorem.)

b = 1260 m

A

B

C

225 m

10.10

0 225 mtan 10.1

b

0(tan 10.1 ) 225 mb

0

225 m =

tan 10.1b

Engineering Fundamentals, By Saeed Moaveni, Third Edition, Copyrighted 2007 7-10

Some Useful Trigonometric Relationships

90o

c

b

a

PYTHAGOREAN THEOREM2 2 2c =a +b

a b asinα= cosα= tanα=

c c bb a b

sinβ= cosβ= tanβ=c c a

For Right Triangles:

cb

a

a b claw of sine: = =

sinA sinB sinC

The sine and cosine law (rule) for any arbitrarily shaped triangle:

C

In order to use the law of sines, you must know

a)two angles and any side, or

b) two sides and a angle opposite one of them.

The law of sines:a) two angles and any side, or

b) two sides and a angle opposite one of them.

A

CB

Example 1: Solve the triangle if C = 28.00, c = 46.8cm, and B = 101.50

sin sin

c b

C B

0 0

46.8

sin 28.0 sin101.5

cm b

0 0(sin28.0 ) (sin 101.5 )(46.8 )b cm

0

0

(sin 101.5 )(46.8 )

sin 28.7 7 c

09 . m

cmb

A = 1800 – B – C = 1800 -101.50 – 28.00 = 50.50

To find side a,

sin sin

c a

C A

0 0

46.8

sin28.0 sin50.5

cm a

0 0(sin28.0 ) (sin50.5 )(46.8 )a cm0

0

(sin50.5 )(46.8 )

(sin28.0 )

cma

076.9

The solution is a = 76.9 cm, b = 97.7 cm, and A = 50.5

2 2 2

2 2 2

2 2 2

law of cosine: a =b +c -2bccosA

b =a +c -2accosB

c =a +b -2abcosC

cb

aIn order to use the law of cosines you must know

a)two sides and the included angle, orb) all three sides.

C

C

Example 2: Solve the triangle if A = 115.20, b = 18.5 m, and c = 21.7m.

b= 18.5m

B

a

A

Ac = 21.7 m

115.20

To find side a,a2 = b2 + c2 – 2bc cos A

a2 = (18.5 m)2 + (21.7 m)2 – 2(18.5 m)(21.7 m)(cos 115.20)

a = 34.0 m

To find angle B, use the law of sines (it requires less computation.)

B = 29.50, C = 35.30, a = 34.0 m

Engineering Fundamentals, By Saeed Moaveni, Third Edition, Copyrighted 2007 7-17

Coordinate Systems are used to locate things with respect to a known origin

Types of coordinate systems:

• Cartesian

• cylindrical

• spherical