TRIGONOMETRY OF RIGHT TRIANGLES

TRIGONOMETRIC RATIOSConsider a right triangle with as one of its acute angles. The trigonometric ratios are defined as follows .

hypotenuse opposite

adjacent

sin = hypotenuse

opposite

cos = hypotenuse

adjacent

tan = adjacentopposite

cot = oppositeadjacent

csc = opposite

hypotenuse

sec = adjacent

hypotenuse

Note: The symbols we used for these ratios are abbreviations for their full names: sine, cosine, tangent, cosecant, secant and cotangent.

RECIPROCAL FUNCTIONSThe following gives the reciprocal relation of the six trigonometric functions.

sin = csc

1

cos = sec

1

tan = cot

1cot =

tan1

csc = sin

1

sec = cos

1

THE PYTHAGOREAN THEOREMThe Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In symbol, using the ABC as shown,

222 bac

ca

b

B

C A

FUNCTIONS OF COMPLIMENTARY ANGLES

ca

b

B

C A

sin A = ca

cos A = cb

tan A = ba

cot A = ab

sec A = cb

csc A = ac

cos B = ca

sin B = cb

cot B = ba

tan B = ab

csc B = cb

sec B = ac

Comparing these formulas for the acute angles A and B, and making use of the fact that A and B are complementary angles (A+B=900), then

FUNCTIONS OF COMPLIMENTARY ANGLES

sin B = sin = cos A

)A90( 0

cos B = cos = sin A )A90( 0

tan B = tan = cot A )A90( 0

cot B = cot = tan A )A90( 0

sec B = sec = csc A )A90( 0

csc B = csc = sec A )A90( 0

The relations may then be expressed by a single statement: Any function of the complement of an angle is equal to the co-function of the angle.

TRIGONOMETRIC FUNCTIONS OF SPECIAL ANGLES 450, 300 AND 600

To find the functions of 450, construct a diagonal in a square of side 1. By Pythagorean Theorem this diagonal has length of .

2450

450

1

1

sin 450 = 22

21

cos 450 = 22

21

tan 450 = 1

csc 450 = 2

sec 450 = 2

cot 450 = 1

2

To find the functions of 300 and 600, take an equilateral triangle of side 2 and draw the bisector of one of the angles. This bisector divides the equilateral triangle into two congruent right triangles whose angles are 300 and 600. By Pythagorean Theorem the length of the altitude is . 3

300

600

3

1

2

sin 300 = 21

cos 300 = 23

csc 300 = 2

tan 300 = 33

31

cot 300 = 3

sec 300 = 3

323

2

sin 600 = 23

cos 600 = 21

tan 600 = 3

cot 600 = 33

31

csc 600 = 3

323

2

sec 600 = 2

EXAMPLE:1. Draw the right triangle whose sides have the

following values, and find the six trigonometric functions of the acute angle A:

2. The point (7, 12) is the endpoint of the terminal side of an angle in standard position. Determine the exact value of the six trigonometric functions of the angle.

a) a=5 , b=12 , c=13

3b) a=1 , b= , c=2

EXAMPLE:3. Find the other five functions of the acute angle A,

given that:

4. Express each of the following in terms of its cofunction:

a) tan A = 43

b) sec A = 2 c) sin A = 22 nmmn2

a) sin 076 b) csc "'0 323580 c) tan )15A( 05. Determine the value of that will satisfy the

ff.:

a) csc = sec 7 )126( 0

b) sin = )54( 0)103sec(

10

EXAMPLE:6. Without the aid of the calculator, evaluate the

following: a) 3 tan2 600 + 2 sin2 300 – cos2 450 b) 5 cot2 450 + 5 tan 450 + sin 300

c) cos2 600 – csc2 300 – sec 300

d) tan 600 + 2 cot 300 – sin 600

e) tan5 450 + cot2 450 – sin4 600