4.3 - RIGHT TRIANGLE

TRIGONOMETRY

IN THIS SECTION, YOU WILL LEARN TO:

Evaluate trigonometric functions of

acute angles Use fundamental trigonometric identities Use a calculator to evaluate trigonometric functions

SIX TRIGONOMETRIC FUNCTIONS OF A RIGHT TRIANGLE:

Use SOH CAH TOA to remember all six of the trigonometric functions:

sin csc

cos sec

tan cot

opp hypA A

hyp opp

adj hypA A

hyp adj

opp adjA A

adj opp

EXAMPLE:

12

Given a right triangle below, find the values

of the six trigonometric functions of    :

12

6

ANSWERS:

Use the Pythagorean Theorem to find the hypotenuse:

6

12

6 5sin

56 5

2 2 26 12 6 5c c

6 5

5sec

2

12 2 5cos

56 5

6 1tan

12 2

csc 5

cot 2

SPECIAL RIGHT TRIANGLES:

45 45 90 and 30 60 90

SIX TRIGONOMETRIC VALUES OF THE SPECIAL RIGHT TRIANGLES:

* Note that certain sine and cosine angles are equal.

1 2 3sin30 sin 45 sin 60

2 2 2

3 2 1cos30 cos45 cos60

2 2 2

3tan30 tan 45 1 tan 60 3

3

COFUNCTIONS OF COMPLEMENTARY ANGLES ARE CONGRUENT:

• Sine-Cosine, Tangent-Cotangent, and Secant-Cosecant are cofunctions.

sin 90 cos cos 90 sin

tan 90 co cot 90 tant

sec 90 csc csc 90 sec

EXAMPLE OF COMPLEMENTARY ANGLES: • Use the given function values and the cofunction relationships to find the indicated trigonometric functions.

3 1Given sin 60 and cos60 ,

2 2

find tan 60 , cos30 and sin30 .

ANSWERS:

3cos30 sin 90 30 sin 60

2

tan 60 cot 90 60 cot30 3

1sin30 cos 90 30 cos60

2

FUNDAMENTAL TRIGONOMETRIC IDENTITIES:

A) Reciprocal Identities:

1 1sin csc

csc sin1 1

cos secsec cos

1 1tan cot

cot tan

FUNDAMENTAL TRIGONOMETRIC IDENTITIES:

B) Quotient Identities:

sintan

cos

coscot

sin

FUNDAMENTAL TRIGONOMETRIC IDENTITIES:

C) Pythagorean Identities:

2 2

2 2

2 2

sin cos 1

tan 1 sec

cot 1 csc

PROOF OF THE SINE PYTHAGOREAN IDENTITY:

Prove: 2 2sin cos 1

x

y

r

2 2

1y x

r r

2 2

2 21

y x

r r

2 2 2x y r

EXAMPLES:

1) Use trigonometric identities to find the

values of  sin and tan given cos 0.24.

ANSWERS:

1) Given : cos 0.24 2 2sin cos 1

22sin 0.24 1 2sin 0.9424

sin 0.97

sin 0.97tan 4.04

cos 0.24

EXAMPLES:

2) Use trigonometric identities to transform the left side of the equation into the right side.

sin coscsc sec

cos sin

ANSWERS:

2 2sin coscsc sec

sin cos sin cos

sin cos2) Prove the identity : csc sec

cos sin

Find a common denominator :

2 2sin coscsc sec

sin cos

1csc sec

sin cos

csc sec csc sec

CALCULATOR EXAMPLES:

1) Use a calculator to evaluate sec 45 2' 3''.

ANSWERS::1) Since most calculators do not have the

second notation, first change this  sec 45 2' 3'' 

    DMS into a decimal form before you find the

value.

1

1Use your calculator as: sec 45.0342

cos 45.0342

or cos 45.0342 1.415x

2 3sec 45 2' 3'' sec 45 sec 45.0342

60 3600

CALCULATOR EXAMPLES:

2)  Using a calculator, find the value in

degrees and radians  of tan 0.1234.

ANSWERS:

 2) Using a calculator, find tan 0.1234.

1

1

Degree mode : tan 0.1234 7.0348

Radian mode : tan 0.1234 0.1228

Check:

Degree mode : tan 7.0348 0.1234

Radian mode : tan 0.1228 0.1234