Calculating Sides of Triangle

A triangle is a closed plane figure with three sides. It has three interior and exterior angles. Usually we represent the vertices of the triangle as A, B and C and the sides opposite to vertex A, B, and C as a, b and c. We can find the unknown sides or angles of the triangle if we know any three measurements of the triangle using trigonometry.

Introduction on calculating sides of the triangle:

The three laws in trigonometry, namely sine law, cosine law and Napier's formula are helpful in finding the unknown sides and angles of the triangle.

Sine law:

In any triangle ABC, a/sin A = b/sin B = c/sin C

Napier's formula:

In any triangle ABC,

1. tan ((A - B)/2) = (a-b)/(a+b) cot (C/2)

2. tan ((B-C)/2) = (b-c)/(b+c) cot (A/2)

3. tan ((C-A)/2) = (c-a)/(c+a)cot (B/2)

Cosine formula:

In any triangle ABC,

1. a2 = b2 + c2 - 2bc cos A

2. b2 = c2 + a2 - 2ca cos B

3. c2 = a2 + b2 - 2ab cos C

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Problems on Calculating Sides of Triangle:

When we do problems on calculating sides of triangle, first we have to make a rough sketch of the triangle.

Ex 1: In a triangle ABC, A = 35°17', C = 45°13', b = 42.1. Solve the unknown sides of the triangle.

Sol:

Step 1: Draw the rough sketch of the triangle.

calculating sides of triangle

Step 2: We know two angles A and C. Let us find the remaining side. Sum of the angles of a triangle is 180°

B = 180 - ( A + C ) = 180 - (35°17' + 45°13') = 99°30'

Step 3: See which formula could be used. Let us use sien formula

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

Step 4: Let us find side a. So a = (bsin A)/sin B

Step 5: Plug the values and find a

a = (42.1 xx sin 35^o 17')/sin (99^o 30')

Step 6: Take logarithm on both sides

log a = log 42.1 + log sin35°17' - log sin99°30'

= 1.6243 + bar1 .7616 - bar1 .9940

= 1.3859 - bar1 .9940

= 1.3919

Step 7: Take the antilogarithm to find a. So, a = 24.65

Step 8: The other unknown side is c.

Step 9:  c = (bsinC)/sin B = (42.1 xx sin 45^o 13')/sin (99^o30')

Step 10: Take log on both sides.

log c = log 42.1 + log sin45°13' - log sin 99°30'

= 1.6243 + bar1 .8511 - bar1 .9940

= 1.4814

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Step 11: To find c, take antilogarithm

c = 30.3

Step 12: Solution: a = 24.65 and c = 30.3

Ex 2: Solve for side c, if a = 5, b = 4 and C = 68°

Step 1: See what formula can be used.

Step 2: Let us use cosine formula. c2 = a2 + b2 - 2ab cos C

Step 3: Plug the values in the formula.

c2 = a2 + b2 - 2ab cos C = 52 + 42 - 2 x 5 x 4cos 68°

= 25 +16 - 40 x 0.3746

= 26.016

c = 5.1
Practice Problems on Calculating Sides of Triangle:

Find the sides of the triangle where a = 1.53 and B = 85 and C = 70

Solution: b = 3.6 and c = 3.4

Solve for side b, if a = 5, c = 4 and B = 75

Solution: b = 5.54 (Rounded off to 2 decimal places)