right triangle trigonometry advanced math 6.2. 2 right triangle 0 < < 90 six trig ratios ...

Right Triangle TrigonometryRight Triangle Trigonometry

Advanced Math 6.2

Advanced Math 6.2 2

Right triangle• 0 < q < 90• Six trig ratios

q

Side adjacent to q

Sid

e o

pposi

te t

o q

hypotenuse

sinopp

hyp

sine

cosadj

hyp

cosine

tanopp

adj

tangent

cschyp

opp

cosecant

sechyp

adj

secant

cotadj

opp

cotangent

Advanced Math 6.2 3

Example• Find the 6 trig

functions of the triangle

q

5

12

13

sin

cos

tan

csc

sec

cot

12

13

5

13

12

5

13

12

13

5

5

12

Advanced Math 6.2 4

If given the angle• Make a triangle• Make the short side 1• Two common triangles

• 45°- 45°- 90°• 30°- 60°- 90°

Advanced Math 6.2 5

45°- 45°- 90° triangle• From geometry

– The two sides opposite equal angles are equal lengths

45°

45°

1

1

• From Pythagorean thrm.– hyp is

2

2

• Find sin, cos, and tan of 45°

Advanced Math 6.2 6

30°- 60°- 90° triangle• Half of an equilateral

triangle

60°

1

2

• From Pythagorean thrm.– opp is

2

3

• Find sin, cos, and tan of 30° and 60°

30°

1

3

Advanced Math 6.2 7

Cofunctions• sin and cos• tan and cot• sec and csc

• Cofunctions of complementary angles are =

2sec30

3

csc60

2

3

Advanced Math 6.2 8

You try• Find sin, cos, and tan of p/3

radians.

Advanced Math 6.2 9

Special angles• Chart on page 462

Advanced Math 6.2 10

TRIG IDENTITIES

1sin

csc

1csc

sin

1

seccos

1cos

sec

1cot

tan

1tan

cot

Reciprocal Identities

sintan

cos

Quotient Identities

coscot

sin

2 21 cot csc

Note

2 21 tan sec

2 2sin cos 1 Pythagorean Identities

22sin sin

sin sin

Advanced Math 6.2 11

SOH CAH TOA

so

h

SOH

ca

h

CAH

to

a

TOA

Advanced Math 6.2 12

Examples• If tan q = 5, find

• cot q• cos q• tan (90° - q)• csc q

Advanced Math 6.2 13

Examples• Use trig identities to show that

cos sec 1

21 sin 1 sin cos

2tan cotcsc

tan

Advanced Math 6.2 14

Using the calculator for trig functions

• Check the mode• Only buttons for sin, cos, and tan

• May need reciprocal identities

Advanced Math 6.2 15

Examples• Evaluate the following using a

calculator. Round to three decimal places.

• sin 16.35°• csc 16.35°• sec 0.75• cos 0.75

Advanced Math 6.2 16

Solving right triangles

60°

18y

Solve for y.

45°

r

20

Solve for r.

Advanced Math 6.2 17

Angles of elevation and depression

observer

observer

object

object

angle of elevation

angle of depression

Advanced Math 6.2 18

Example• A 10´ ladder leans against the side

of a house. The ladder makes an angle of 60° with the ground. How far up the side of the house does the ladder reach?