Trigonometry

T R I G O N O M E T R Y m . e s t e v e z

a . m a r r a n z i n i

GEOMETRY CLASS

Released June 2020

Definition, father and origin

New digital edition

Trigonometric functions

available now!

Ads, graphs and more

TABLE OF CONTENTS definition / ads / functions / origin / graphs & charts

03

04

ad

definition & origin

11

right triangle similarity

12

ad

13

geometric mean

14

complementary angles & inverse ratios

15

05

06

triangle trigonometry

pythagorean theorem

area & solving of a triangle

16

laws

17

real life

18

applications

19

back cover

June 2020 trigonometry

07

08

ad

news / babylonian trigonometry

09

10

game

theorems

PAGEÂ 3 TRIGONOMETRY | 2020

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W H A T

I S

T R I G O N O M E T R Y ?

Origin

Definition Trigonometry is a branch

O r i g i n

of mathematics that studies relationships between side lengths and angles of triangles.

The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.

WHERE

DID

IT

COME

FROM? The word trigonometry comes Hipparchus was a Greek

from the Greek words

astronomer who lived between 190-120 B.C. He is considered

trigonon (“triangle”) and

the father of trigonometry, a

metron (“to measure”).

studies the angles of sides of

branch of mathematics which triangles.

PAGE 5 TRIGONOMETRY | 2020

Triangle TRIGONOMETRY B

sin (<A) = opposite hypotenuse hy po te

cos (<A) = adjacent se

opposite

nu

hypotenuse tan (<A) = opposite adjacent

C

adjacent

A

T-I-Y

“Can you imagine young

TRY IT YOURSELF

study of trigonometry for the

people nowadays making a fun of it? Well I did”

R

-Clyde Tombaugh

28

sin (<S) =

53

cos (<S) =

T

45

S

tan (<S) =

"Pythagorean theorem"

a

c

a2 + b2 = c2

b

It states that in a right triangle, the sum of the squares of the two sides are equal to the longest side, which is called the hypotenuse

DEFINITION

REAL LIFE

It can be used in architecture, navigation, surveying, etc.

2

EXAMPLE

5

3

4 PAGE 6 TRIGONOMETRY | 2020

2

3 +4 =5

2

9 + 16 = 25

PAGE 7 TRIGONOMETRY | 2020

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Breaking

news

Ancient Babylonian trigonometry was 'superior' to modern day versionThey also beat the Ancient Greeks to it, according to Australian academics

BABYLONIAN TRIGONOMETRY The ancient Babylonians who lived from about 4,000BCE in what is now Iraq had a long forgotten understanding of right angled triangles that was much simpler and more accurate than the conventional trigonometry we are taught in schools.

PAGE 8 TRIGONOMETRY | 2020

w o r d WORD s e a r c h SEARCH 01

trigonometry

02

triangle

03

angle

04

cosine

05

tangent

e k d c j w i f c n c o s i n e a

06

sine

n e j s t w t k c n s u c l g d e

07

adjacent

a j v p v l v f u h b w q s i n e n k o m n y h i r d s a t u b o g h y t a n g e n t e l r m b c l k c o w c u b w j d i e u o p

b s u d l e o w u n g d g j j m b v c x z l k j h g f d s a a o i u y t r e w q m n b v c c z l k j h g f d s a p o i u y t e p w q p l m o k n i j b u h v n g c t r i a n g l e s a q w q t e r t r i g o n o m e t r y h j k l z x c v b n m z a q x s w c d e v f r b g t n h t y u j m k

T H E O R E M S PYTHAGOREAN INEQUALITIES

If c² > a² + b² then the triangle is

If c² < a² + b² then the triangle is

obtuse

acute

c

a

c² > a² + b²

c

a

obtuse

c² < a² + b² acute

b

b

45°- 45°- 90° TRIANGLE

30°- 60°- 90° TRIANGLE

the hypothenuse is √2 times as long as each leg

the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.

45°

60°

x

x√2

45° x

x

2x

30° x√3

Right triangle similarity T H E O R E M

If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.

B

D

ABC ~ ACD ~ CBD

C

A

PAGEÂ 12 TRIGONOMETRY | 2020

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GEOMETRIC MEAN

The geometric mean of two positive numbers, A and B, is the positive number x that satisfies a/x = x/b. So, x² and x=√ab

GEOMETRIC MEAN ALTITUDE THEOREM In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments.The length of the altitude is the geometric mean of the lengths of the two segments of the hypotenuse.

altitude rule:

h²=x . y

GEOMETRIC MEAN LEG THEOREM In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments.The lengths of each leg of the right angle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

leg rule:

a²=y.c b²=x.c

14

SINE AND COSINE OF COMPLEMENTARY ANGLES The sine of an acute angle is equal to the cosine of its complement. The cosine of an acute angle is equal to the sine of its complement.

Let A and B be complementary angles. Then the following statements are true

Sin A= cos(90˚-A) =cos B

Sin B= Cos(90˚-B) =cos A

Cos A= Sin(90˚-B) = sin B

Cos B= sin(90˚-B) =sin A

INVERSE TRIGONOMETRIC RATIOS B

Let <A be acute Inverse tangent if tan A= x, then x= mA

tan^(-1)

BC/AC=m<A

Inverse Sine if sin A= y, then y= m<A

sin^(-1)

BC/AB=m<A

Inverse Cosine if cos A= z, then z= m<A

A

C

cos^(-1)

AC/AB=m<A

PAGE 15 TRIGONOMETRY | 2020

Solving A RIGHT TRIANGLE

“Trigonometry must be a

To solve a right triangle, you have to find all unknown side lengths and

beach-person. You see the 'tan'

angle measures. You can solve a

it got?”

right triangle when you have either

-Lunostellar

of the following:

· Two side lengths · One side length and the measure of one acute angle

C

Area

OF A TRIANGLE b

a

The area of any triangle is given by onehalf the product of the lengths of two sides times the sine o their included

A

c

B

angle. For ABC shown there are three ways to calculate the area:

Area= 1/2bc (sin A) Area= 1/2ac (sin B) Area= 1/2ab (sin C)

PAGE 16 TRIGONOMETRY | 2020

Law of sine This law can be written in either of the following forms for ABC with side length a, b and c.

C

b

A/sin A =b/sin B A/sin A =b/sin B

a

sin A/a=sin B/b=sin C/c A

c

B

Law of cosine If ABC has sides of length a, b and c, as shown, then the following are true.

C a² = b² + c² - 2bc cos A b

a

b² = a² + c² - 2ac cos B

c² = a² + b² - 2ab cos C A

c

B

REAL LIFE APPLICATION p y t h a g o r e a n

t h e o r e m

These stair steps form a right triangle, as shown.

Use the Pythagorean theorem to aproximate the distance between the the two sides

x

16.7 ft

8.9 ft

18

Real life APPLICATION

Imagine you are skiing on a mountain with an altitude of 1200 feet. The angle of depression is 21°.

Find the distance x you ski down the mountain to the nearest foot

21° x ft

1200 ft

â&#x20AC;?

Life is like a trigonometry problem, when we start it, every step becomes difficult.

At many places we think let's quit, do another one, but at the end we proudly say 'hence prove' -Gaurav Sharma

TRIGONOMETRY