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

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

following values, and find the six trigonometric functions of the acute angle A: a) a=5 , b=12 , c=13

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

following values, and find the six trigonometric functions of the acute angle A: 3b) a=1 , b= , c=2

EXAMPLE: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.

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

given that:

a) tan A = 43

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

given that:

b) sec A = 2

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

given that:

c) sin A = 22 nmmn2

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.

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

cofunction:a) sin 076 b) csc "'0 323580 c) tan )15A( 0

EXAMPLE:5. Determine the value of that will satisfy the

ff.:

a) csc = sec 7 )126( 0

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

10

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: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

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

following: a) 3 tan2 600 + 2 sin2 300 – cos2 450

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

following: b) 5 cot2 450 + 5 tan 450 + sin 300

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

following: c) cos2 600 – csc2 300 – sec 300

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

following: d) tan 600 + 2 cot 300 – sin 600

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

following: e) tan5 450 + cot2 450 – sin4 600

Find….

1.sin 32 o =

2.cos 81 o =

3.tan 18 o =

4.sec 58 o =

5.cot 78 o =

IF sin = 0.2588 find IF cos = 0.3746 find IF tan = 4.011 find

Use Trigonometry To Find Angles

A

B

C19 .ซม

34

ac

a …………………..c …………………..

Use trigonometric about special right triangles to find the value of x and y.

Use trigonometric about special right triangles to find the value of x and y.

Find the missing lengths

Find the missing lengths

*The angle between the HORIZONTAL and a line of sight is called an angle of elevation or an angle of depression

Trigonometric Word Problems

A 20-foot ladder is leaning against a wall. The base of the ladder is 10 feet from the wall. What angle does the ladder make with the ground

10

?

20

Cos A

Cos A

A = 60°

How tall is a bridge if a 6-foot tall person standing 100 feet away can see the top of the bridge at an angle of 60 degrees to the horizon?

100 6

60°

A hot air balloon is flying at an altitude of 1500 m. The angle of depression from the balloon to a landmark on the ground is 30º.

a) What is the balloon’s horizontal distance to the landmark, to the nearest metre?

b) What is the balloon’s direct distance to the landmark, to the nearest metre?

Two buildings are 30 m apart. The angle from the top of the shortest building to the top of the taller building is 30°. The angle from the top of the shorter building to the base of the taller building is 45°. What is the height of the taller building to the nearest metre?