The Pythagorean TheoremThe Pythagorean Theorem

From From The Simpson’s The Simpson’s Homer copies the Homer copies the Scarecrow on Scarecrow on The Wizard of OzThe Wizard of Oz..

The Pythagorean TheoremThe Pythagorean Theorem

The sum of the square of the two sides of a The sum of the square of the two sides of a right triangle right triangle is equal to is equal to the square of the hypotenuse.the square of the hypotenuse.

The sum of the square of the two sides of a The sum of the square of the two sides of a right triangle right triangle is equal to the square of the hypotenuse.is equal to the square of the hypotenuse.

An example of the Pythagorean TheoremAn example of the Pythagorean Theorem

• Let the adjacent side of the right triangle be 10 and the Let the adjacent side of the right triangle be 10 and the opposite side be 12. Find the hypotenuse of the right triangle.opposite side be 12. Find the hypotenuse of the right triangle.

2 2 2

2

2

10 12

100 144

244

244

2 61

r

r

r

r

r

12

10

r

Now you try it!Now you try it!

• Let the adjacent side of the right triangle be 12 and the Let the adjacent side of the right triangle be 12 and the hypotenuse be 13. Find the opposite side of the right triangle.hypotenuse be 13. Find the opposite side of the right triangle.

2 2 2

2

2

12 13

144 169

25

25

5

y

y

y

y

y

y

12

13

Definition of the Trig. Definition of the Trig. FunctionsFunctions

cos sec

sin csc

tan cot

adj hypx rhyp r adj x

opp y hyp rhyp r opp y

opp y adj xadj x opp y

An example of the Trigonometric FunctionsAn example of the Trigonometric Functions

• Let the adjacent side of the right triangle be 10 and the Let the adjacent side of the right triangle be 10 and the opposite side be 12. Find all the trigonometric functions of the opposite side be 12. Find all the trigonometric functions of the given angle theta.given angle theta.

Previously using the pythagorean theorem,

we know 2 61 .r

12

10

r

12

10

2 61

2 61 6110 510 52 61 61

2 61 6161212 62 61 61

6 101210 5 1

Thus, using the triangle above and the

right triangle trigonometric ratios, we have

cos sec

sin csc

tan cot

adj hyphyp adj

opp hyphyp opp

opp adjadj opp

52 6

Now you try it!Now you try it!• Let the adjacent side of the right triangle be 12 and the hypotenuse Let the adjacent side of the right triangle be 12 and the hypotenuse

be 13. Find all the trigonometric functions of the given angle theta.be 13. Find all the trigonometric functions of the given angle theta.

y

12

13

Again, by the pythagorean theorem,

you should have found, 5.y

5

12

13

131213 12

5 1313 5

5 1212 5

Thus, using the triangle above and the

right triangle trigonometric ratios, we have

cos sec

sin csc

tan cot

adj hyphyp adj

opp hyphyp opp

opp adjadj opp

Special Angles and Trig. Special Angles and Trig. FunctionsFunctions•The 30°-60°-90° triangle:

2 2 2

2

Using the pythagorean theorem

on either smaller triangle,

1 2

3

3

3

y

y

y

y

2 2

1

60°

30°

y

Thus the 30°-60°-90° triangle sides are in the ratio of 1 - - 2.3

Trig. Functions of 30°-60°-90° Trig. Functions of 30°-60°-90° triangles.triangles.

3

2 2

1

60°

30°

3 12 2

312 2

3113

Using the triangle above and the right triangle trigonometric ratios

cos 30 cos 60

sin 30 sin 60

tan 30 tan 60 3

adj adjhyp hyp

opp opphyp hyp

opp oppadj adj

Special Angles and Trig. Special Angles and Trig. FunctionsFunctions•The 45°-45°-90° triangle:

2 2 2

2

Using the pythagorean theorem

on the isosceles triangle.

1 1

2

2

2

r

r

r

r

Thus the 45°-45°-90° triangle sides are in the ratio of 1 - 1 - .2

1

1

45°

45°r

Trig. Functions of 45°-45°-90° Trig. Functions of 45°-45°-90° triangles.triangles.

12

12

11

Using the triangle shown and

the right triangle trigonometric

ratios,

cos 45

sin 45

tan 45 1

adjhyp

opphyp

oppadj

1

1

45°

45°

2

The Unit CircleThe Unit Circle

1

1

Since 1 on the unit circle, the trig. functions

are defined as follows:

cos sec

sin csc

tan cot

adj hyphyp adj x

opp hyphyp opp y

opp y adj xadj x opp y

r

x

y

The Unit Circle: The Unit Circle: cos ,sin