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ENGINEERING MATHEMATICS-I

DIPLOMA IN MECHANICAL ENGINEERING

1st SEMESTER

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ANANYA MISHRA

ASST. PROFESSOR

GANDHI INSTITUTE FOR EDUCATION AND TECHNOLOGY, BANIATANGI, BHUBANESWAR

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Right Triangle Trigonometry

Trigonometry is based upon ratios of the sides of right triangles.

The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle.

θ

opp

hyp

adj

The sides of the right triangle are:

 the side opposite the acute angle ,

 the side adjacent to the acute angle ,

 and the hypotenuse of the right triangle.

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A

A

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The hypotenuse is the longest side and is always opposite the right angle.

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The opposite and adjacent sides refer to another angle, other than the 90o.

Right Triangle Trigonometry

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S O H C A H T O A

The trigonometric functions are:

sine, cosine, tangent, cotangent, secant, and cosecant.

opp

adj

hyp

θ

sin = cos = tan =�

csc = sec = cot = �

opp

hyp

adj

hyp

hyp

adj

adj

opp

opp

adj

Trigonometric Ratios

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Finding an angle from a triangle

To find a missing angle from a right-angled triangle we need to know two of the sides of the triangle.

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We can then choose the appropriate ratio, sin, cos or tan and use the calculator to identify the angle from the decimal value of the ratio.

Find angle C

  1. Identify/label the names of the sides.

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b) Choose the ratio that contains BOTH of the letters.

14 cm

6 cm

C

1.

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C = cos-1 (0.4286)

C = 64.6o

14 cm

6 cm

C

1.

h

a

We have been given the adjacent and hypotenuse so we use COSINE:

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Cos A =

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Cos A =

Cos C =

Cos C = 0.4286

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Find angle x

2.

8 cm

3 cm

x

a

o

Given adj and opp

need to use tan:

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Tan A =

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x = tan-1 (2.6667)

x = 69.4o

Tan A =

Tan x =

Tan x = 2.6667

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Cos 30 x 7 = k

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6.1 cm = k

7 cm

k

30o

3.

We have been given the adj and hyp so we use COSINE:

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Cos A =

​

Cos A =

Cos 30 =

Finding a side from a triangle

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Tan 50 x 4 = r

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4.8 cm = r

4 cm

r

50o

4.

Tan A =

Tan 50 =

We have been given the opp and adj so we use TAN:

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Tan A =

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45°-45°-90° Triangle Theorem

  • In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg.

x√2

45°

45°

Hypotenuse = √2 * leg

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30°-60°-90° Triangle Theorem

  • In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.

x√3

60°

30°

Hypotenuse = 2 ∙ shorter leg

Longer leg = √3 ∙ shorter leg

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Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle

  • Find the value of x
  • By the Triangle Sum Theorem, the measure of the third angle is 45°. The triangle is a 45°-45°-90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg.

3

3

x

45°

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Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle

Hypotenuse = √2 ∙ leg

x = √2 ∙ 3

x = 3√2

3

3

x

45°

45°-45°-90° Triangle Theorem

Substitute values

Simplify

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Ex. 3: Finding side lengths in a 30°-60°-90° Triangle

  • Find the values of s and t.
  • Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg.

30°

60°

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Ex. 3: Side lengths in a 30°-60°-90° Triangle

Statement:

Longer leg = √3 ∙ shorter leg

5 = √3 ∙ s

Reasons:

30°-60°-90° Triangle Theorem

5

√3

√3s

√3

=

5

√3

s

=

5

√3

s

=

√3

√3

5√3

3

s

=

Substitute values

Divide each side by √3

Simplify

Multiply numerator and denominator by √3

Simplify

30°

60°

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The length t of the hypotenuse is twice the length s of the shorter leg.

Statement:

Hypotenuse = 2 ∙ shorter leg

Reasons:

30°-60°-90° Triangle Theorem

t

2 ∙

5√3

3

=

Substitute values

Simplify

30°

60°

t

10√3

3

=