By Nittaya Noinan

Kanchanapisekwittayalai phetchabun school

Grade 10

Trigonometry

Trigonometry is derived from Greek words trigonon (three angles) and metron ( measure).

Trigonometry is the branch of mathematics which deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees

Triangles on a sphere are also studied, in spherical trigonometry.

Trigonometry specifically deals with the relationships between the sides and the angles of triangles, that is, on the trigonometric functions, and with calculations based on these functions.

Trigonometry

History

The origins of trigonometry can be traced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley, more than 4000 years ago.

Some experts believe that trigonometry was originally invented to calculate sundials, a traditional exercise in the oldest books

The first recorded use of trigonometry came from the Hellenistic mathematician Hipparchus circa 150 BC, who compiled a trigonometric table using the sine for solving triangles.

The Sulba Sutras written in India, between 800 BC and 500 BC, correctly compute the sine of π/4 (45°) as 1/√2 in a procedure for circling the square (the opposite of squaring the circle).

Many ancient mathematicians like Aryabhata, Brahmagupta,Ibn Yunus and Al-Kashi made significant contributions in this field(trigonometry).

Right Triangle

A triangle in which one angle is equal to 90 is called right triangle.

The side opposite to the right angle is known as hypotenuse.

AB is the hypotenuse

The other two sides are known as legs.

AC and BC are the legs

Trigonometry deals with Right Triangles

Pythagoras Theorem

In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of areas of the squares whose sides are the two legs.

In the figureAB2 = BC2 + AC2

Trigonometry (Trigonometry (三角幾何三角幾何 )) means means “Triangle” and “Measurement”“Triangle” and “Measurement”

Introduction Trigonometric Introduction Trigonometric RatiosRatios

In F.2 we concentrated on right angle trianglesIn F.2 we concentrated on right angle triangles.

Unit CircleUnit Circle

A Unit Circle Is a Circle With Radius Equals to 1 Unit.(We Always Choose Origin As Its centre)

1 units

x

Y

Trigonometry

Trigonometry is the branch of mathematics that looks at the relationship between the length of the sides and the angles in a right angled triangle.It helps us calculate the length of unknown sides, without drawing the triangle, and it also helps us calculate the size of an unknown angle without having to measure it.Let us look at a right angle triangle.

Trigonometry

O F

C

Hypotenuse

x0

Opposite

Adjacent

O F

C

Hypotenusey0

Opposite

Adjacent

The ‘Hypotenuse’ is always opposite the right angle

The ‘Opposite’ is always opposite the angle under investigation.

The ‘Adjacent’ is always alongside the angle under investigation.

Trigonometry

12

6

3

O F

C

CFOF

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Let us look at the ratio or fraction of the opposite over the adjacent.

Let us now split this triangle up.

270

Trigonometry

12

4 2

23

O E F

B

C

CFOF

36

BEOE

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12

Let us look at the ratio or fraction of the opposite over the adjacent

Let us split it again.

270

Adjacent , Opposite Side and Adjacent , Opposite Side and Hypotenuse of a Right Angle Hypotenuse of a Right Angle

TriangleTriangle..

Adjacent side

Opposite side

hypotenuse

hypotenuse

Adjacent side

Opposite side

There are 3 kinds of trigonometric ratios we will learn.

sine ratio

cosine ratio

tangent ratio

Three Types Trigonometric Three Types Trigonometric RatiosRatios

Sine RatiosSine Ratios

Definition of Sine Ratio. Application of Sine Ratio.

Definition of Sine Ratio.

1

If the hypotenuse equals to 1

Sin = Opposite sides

Definition of Sine Ratio.

For any right-angled triangle

Sin = Opposite side

hypotenuses

Exercise 1

4

7

In the figure, find sin

Sin = Opposite Side

hypotenuses

= 47

= 34.85 (corr to 2 d.p.)

Exercise 2

11

In the figure, find y

Sin35 = Opposite Side

hypotenuses

y11

y = 6.31 (corr to 2.d.p.)

3535°°

y

Sin35 =

y = 11 sin35

Cosine RatiosCosine Ratios

Definition of Cosine. Relation of Cosine to the sides of right

angle triangle.

Definition of Cosine Ratio.

1

If the hypotenuse equals to 1

Cos = Adjacent Side

Definition of Cosine Ratio.

For any right-angled triangle

Cos = hypotenuses

Adjacent Side

Exercise 3

3

8

In the figure, find cos

cos = adjacent Side

hypotenuses

= 38

= 67.98 (corr to 2 d.p.)

Exercise 4

6

In the figure, find x

Cos 42 = Adjacent Side

hypotenuses

6x

x = 8.07 (corr to 2.d.p.)

4242°°

x

Cos 42 =

x =

6Cos 42

Tangent RatiosTangent Ratios

Definition of Tangent. Relation of Tangent to the sides of

right angle triangle.

Definition of Tangent Ratio.

For any right-angled triangle

tan = Adjacent Side

Opposite Side

Exercise 5

3

5

In the figure, find tan

tan = adjacent Side

Opposite side

= 35

= 78.69 (corr to 2 d.p.)

Exercise 6

z

5

In the figure, find z

tan 22 = adjacent Side

Opposite side

5

z

z = 12.38 (corr to 2 d.p.)

2222

tan 22 =

5

tan 22z =

ConclusionConclusion

hypotenuse

side oppositesin

hypotenuse

sidedjacent acos

sidedjacent a

side oppositetan

Make Sure that the

triangle is right-angled

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Trigonometric ratios

Sine(sin) opposite side/hypotenuseCosine(cos) adjacent side/hypotenuse Tangent(tan) opposite side/adjacent side Cosecant(cosec) hypotenuse/opposite sideSecant(sec) hypotenuse/adjacent sideCotangent(cot) adjacent side/opposite side

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Values of trigonometric function of Angle A

sin = a/c

cos = b/c

tan = a/b

cosec = c/a

sec = c/b

cot = b/a

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Values of Trigonometric function0 30 45 60 90

Sine 0 0.5 1/2 3/2 1

Cosine 1 3/2 1/2 0.5 0

Tangent 0 1/ 3 1 3 Not defined

Cosecant Not defined 2 2 2/ 3 1

Secant 1 2/ 3 2 2 Not defined

Cotangent Not defined 3 1 1/ 3 0

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Calculator

This Calculates the values of trigonometric functions of different angles.

First Enter whether you want to enter the angle in radians or in degrees. Radian gives a bit more accurate value than Degree.

Then Enter the required trigonometric function in the format given below:

Enter 1 for sin. Enter 2 for cosine. Enter 3 for tangent. Enter 4 for cosecant. Enter 5 for secant. Enter 6 for cotangent. Then enter the magnitude of angle.

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Trigonometric identities sin2A + cos2A = 1 1 + tan2A = sec2A 1 + cot2A = cosec2A sin(A+B) = sinAcosB + cosAsin B cos(A+B) = cosAcosB – sinAsinB tan(A+B) = (tanA+tanB)/(1 – tanAtan B) sin(A-B) = sinAcosB – cosAsinB cos(A-B)=cosAcosB+sinAsinB tan(A-B)=(tanA-tanB)(1+tanAtanB) sin2A =2sinAcosA cos2A=cos2A - sin2A tan2A=2tanA/(1-tan2A) sin(A/2) = ±{(1-cosA)/2} Cos(A/2)= ±{(1+cosA)/2} Tan(A/2)= ±{(1-cosA)/(1+cosA)}

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Relation between different Trigonometric Identities

Sine Cosine Tangent Cosecant Secant Cotangent

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Angles of Elevation and Depression

Line of sight: The line from our eyes to the object, we are viewing.

Angle of Elevation:The angle through which our eyes move upwards to see an object above us.

Angle of depression:The angle through which our eyes move downwards to see an object below us.

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Problem solved using trigonometric ratios

CLICK HERE!

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Applications of Trigonometry This field of mathematics can be applied in astronomy,navigation,

music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.

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Derivations Most Derivations heavily rely on

Trigonometry.

Click the hyperlinks to view the derivation A few such derivations are given below:- Parallelogram law of addition of vectors. Centripetal Acceleration. Lens Formula Variation of Acceleration due to gravity due

to rotation of earth. Finding angle between resultant and the

vector.

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Applications of Trigonometry in Astronomy

Since ancient times trigonometry was used in astronomy. The technique of triangulation is used to measure the distance to nearby stars. In 240 B.C., a mathematician named Eratosthenes discovered the radius of the

Earth using trigonometry and geometry. In 2001, a group of European astronomers did an experiment that started in 1997

about the distance of Venus from the Sun. Venus was about 105,000,000 kilometers away from the Sun .

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Application of Trigonometry in Architecture

Many modern buildings have beautifully curved surfaces. Making these curves out of steel, stone, concrete or glass is

extremely difficult, if not impossible. One way around to address this problem is to piece the

surface together out of many flat panels, each sitting at an angle to the one next to it, so that all together they create what looks like a curved surface.

The more regular these shapes, the easier the building process.

Regular flat shapes like squares, pentagons and hexagons, can be made out of triangles, and so trigonometry plays an important role in architecture.

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Waves

The graphs of the functions sin(x) and cos(x) look like waves. Sound travels in waves, although these are not necessarily as regular as those of the sine and cosine functions.

However, a few hundred years ago, mathematicians realized that any wave at all is made up of sine and cosine waves. This fact lies at the heart of computer music.

Since a computer cannot listen to music as we do, the only way to get music into a computer is to represent it mathematically by its constituent sound waves.

This is why sound engineers, those who research and develop the newest advances in computer music technology, and sometimes even composers have to understand the basic laws of trigonometry.

Waves move across the oceans, earthquakes produce shock waves and light can be thought of as traveling in waves. This is why trigonometry is also used in oceanography, seismology, optics and many other fields like meteorology and the physical sciences.

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Digital Imaging In theory, the computer needs an infinite amount of information to do

this: it needs to know the precise location and colour of each of the infinitely many points on the image to be produced. In practice, this is of course impossible, a computer can only store a finite amount of information.

To make the image as detailed and accurate as possible, computer graphic designers resort to a technique called triangulation.

As in the architecture example given, they approximate the image by a large number of triangles, so the computer only needs to store a finite amount of data.

The edges of these triangles form what looks like a wire frame of the object in the image. Using this wire frame, it is also possible to make the object move realistically.

Digital imaging is also used extensively in medicine, for example in CAT and MRI scans. Again, triangulation is used to build accurate images from a finite amount of information.

It is also used to build "maps" of things like tumors, which help decide how x-rays should be fired at it in order to destroy it.

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Conclusion

Trigonometry is a branch of Mathematics with several important and useful applications.

Hence it attracts more and more research with several theories published year after year

Thank You

END