Sine Cosine Formula

In trigonometry, among the six ratios, sine and cosine are two ratios.  In a right angled triangle ABC,

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Sin x = [BC]/[AC] .

This can be also written as Sinx = [Opp]/[Hyp] .

Cos x = [AB]/[AC] .

This can be also written as Cosx = [Adj]/[Hyp] .


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Besides this, we one more formula in sine as well as for cosine which we call it as sine rule and cosine rule respectively. They are as follows:

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1. Sine rule: a / [sinA] = b/[sinB] = c/[sinC] .

2. Cosine rule:

a2 = b2 + c2 – 2bc cosA.

b2 = c2 + a2 – 2ca cosB.

c2 = a2 + b2 – 2ab cosC.

Now let us use these formula’s to find the values of the unknown.
Example Problems on Sine and Cosine Formula.

Ex 1: Find the value sinx and cosx from the given diagram.

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Solution: By pythagoral theorem,

Hypotenuse = sqrt [5^2 + 12^2]  = sqrt169 = 13.

Therefore Sinx = 5/13 .

Cosx = 12/13 .

Ex 2: Find the value of A, B and b in the given triangle.

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Solution: By using sine rule, we have:

sinA/12  = sin30/ 8 implies sinA = 1/2 xx 12 / 8 = 6/8 = 3/4 .

implies A = 48.60.

Therefore B = 180 – (48.6 + 30) = 101.40.

Also by using sine rule again,

b / [sin 101.4] = 12 /sin 48.6

Therefore b = [12 xx sin 101.4] / [sin 48.6] = 15.7.


Ex 3: Find the value of the unknown in the given triangle:

sincosin5

Solution: Let us use cosine rule to find the unknowns in this triangle.

Given: B = 490 , a = 12, c = 15.

Therefore b2 = a2 + c2 – 2ac cosB.

= 122 + 152 – 2 (12) (15) cos49.

= 132.82.

Also cosA = [a^2 ** (b^2 + c^2)] / [2bc] .

Implies cosA = [12^2 ** ( 11.52^2 + 15^2 )] / [2 (11.52) (15)] .

Implies A = 1280.1

Therefore C = 180 – (49 + 128.1) = 2.90.
Practice Problems on Sine and Cosine Formula:

1. Find the value of the unknown in the given triangle [one decimal place]

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[Ans: a = 7]

2. Find the value of the unknown in the given triangle: [one decimal place]

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[Ans: a = 7.7 cm]