Engineering Mathematics Chapter 2: Trigonometry

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Engineering Mathematics Chapter 2:

Trigonometry

Muhammad Adi Haziq Bin Mohd Basir | DKA1C | 11DKA23F1018 Muhammad Iz’aan Fahmi Bin Kamarudzaman | DKA1C |

11DKA23F1006

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Introduction of Trigonometry

• Trigonometry is one of the most important branches in mathematics.

• The word trigonometry is formed by clubbing words 'Trigonon' and 'Metron' which means triangle and measure respectively.

• It is the study of the relation between the sides and angles of a right-angled triangle.

• It thus helps in finding the measure of unknown dimensions of a

right-angled triangle using formulas and identities based on this relationship.

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Trigonometry basics

Trigonometry basics deal with the measurement of angles and problems related to angles.

There are three basic functions in trigonometry:

SINE COSINE TANGENT

Here three basic ratios or functions can be used to derive other important trigonometric functions:

COSECANT SECANT COTANGENT

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Sine =

Cosine =

Tangent =

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Sine,

cosine &

tangent

Triangles have been a very important part of geometry so far, and the important characteristics of triangles are that they have three sides (as indicated in the

name), and 3 angles that add up to 180. It is important to certain characteristics of the angles of right triangles, and this is where sine, cosine, and tangent come into play.

How do we calculate sine, cosine, and tangent?

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CALCULATIO N

First let’s understand our main

component

before calculating Sine/Cosine/Tangent.

A

B C

We have to identify the opposite, adjacent and

hypotenuse sides. To illustrate, let’s use one angle of the triangle.In a right triangle, the hypotenuse is the

longest side, an "opposite" side is the one across from a given angle, and an

"adjacent" side is next to a given angle.

Hypotenuse Opposite

Adjacent

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Applications of Trigonometry

Throughout history, trigonometry has been applied in areas such as architecture, celestial mechanics, surveying, etc. Its applications include in:

• Various fields like oceanography, seismology, meteorology,

physical sciences, astronomy, acoustics, navigation, electronics, and many more.

• It is also helpful to find the distance of long rivers, measure the height of the mountain, etc.

• Spherical trigonometry has been used for locating solar, lunar, and stellar positions.

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Question :

A surveyor measures the angle of elevation of the top of a perpendicular building as 19°. He move 120m nearer to the building and find the angle of elevation is now 47°.

Determine the height of the building.

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Illustration

Surveyor

4

Building

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Illustration

A surveyor measures the angle of elevation of the top of a perpendicular building as 19°.

19°

He move 120m nearer to the building 120m

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Illustration

A surveyor measures the angle of elevation of the top of a perpendicular building as 19°.

19°

He move 120m nearer to the building 120m

and find the angle of elevation is now 47°.

47°

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Sine Formula

The Law of Sines (or Sine Rule) is very useful for solving triangles:

A, B and C are sides.

A, B and C are angles.

C A

B C

A B

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19°

120m

47°

We can use Sine formula to determine the length of the side triangle.

A B

C

D

Let’s mark the ABC triangle angles first.

In this scenario, we have to find the length of BC, then we can proceed with our

building height calculation, using the length of DC.

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19

°

120 m

47°

A B

C

D

Let’s mark the sides of the triangle ABC B A

C

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19

°

120 m

47°

A B

C

D

Before using the formula, we need to calculate the angle of C.

B A C

This is because we already have the value

of the length of AB thus finding the angle of C can make it applicable into our equation.

By finding both value, we can determine the length of BC which is illustrated as A.

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19

°

120 m

47°

A B

C

D

B A C

To determine the angle of ∠BAC, we need to calculate the angle of

∠ABC, first we knew that ABD are straight line, So here, we can determine our angle:

∠ABC – ∠BCD = 180° – 47°

∠BCD = 133°

133°

∠BCA:

180° - (CAB + ABC) = 180° – (19° + 133°)

∠BCA = 28°.

28°

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19

°

120 m

47°

A B

C

D

B A C

Now, we can apply our angle calculation into sine formula.

133°

28°

Here, we can use 120m as value of C, while Sin C, 28°

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19

°

120 m

47°

A B

C

D

B A C

133°

28°

For , we only got an angle of A,

19° to be applied at Sin A. While the length of BC (A) is to be evaluate.

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19

°

120 m

47°

A B

C

D

B A C

133°

28°

For , we only got an angle of A,

19° to be applied at Sin A. While the length of BC (A) is to be evaluate.

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19

°

120 m

47°

A B

C

D

B A C

133°

28°

Let’s rearrange the equation, then we solve it.

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19

°

120 m

47°

A B

C

D

B A C

133°

28°

Let’s rearrange the equation, then we solve it.

� = 83.22 �

Value of the length of BC.

83.22m

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19

°

120 m

47°

A B

C

D

B A C

133°

28°

Now we can finally find the value of height of the

building.

However as we know, we have to find one angle with it’s value.

83.22m

In this scenario,

let’s use ∠BDC as our angle which is 90.

We assume that the height of building is the length of CD. Let’s find both angles and sides for our equation.

D B

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19

°

120 m

47°

A B

C

D

B A C

133°

28°

83.22m B D

Apply these value to our equation.

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19

°

120 m

47°

A B

C

D

B A C

133°

28°

83.22m B D

Rearrange our equation to get our answer..

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19

°

120 m

47°

A B

C

D

B A C

133°

28°

83.22m B D

B

Rearrange our equation to get our answer..

B = 60.86m

Here, we finally found that the height of the building is 60.86m.

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Engineering Mathematics Chapter 2: Trigonometry