Sine and Cosine Rule

 Sine and Cosine Rule 

 

How useful are sine and cosine rule

Everyone thinks that what is the use sine and cosine rule so we can understand in this blog that how sine and cosine rule helps in calculating the big calculation. Now let's start- 

We can easily find the sides of a right triangle if one side  and one angle is given in the question or in triangle.

But if the triangle is not right triangle then What's you do???

Now, we use sine and cosine rule to find the side of a triangle if the triangle is not right angled. 

So let's start sine and cosine rule 

Sine Rule 

In∆ABD 

SinB=AD/AB 

AD=cSinB —1 

In ∆ACD 

SinC=AD/AC 

AD=bSinC —2 

From equation 1 and 2, we get 

cSinB=bSinC 

➡️ c/SinC=b/SinB —3 

And, 

  In ∆BAE 

SinA=BE/AB 

BE=cSinA —4

In ∆BCE 

SinC=BE/BC 

BE=aSinC —5 

From equation 4 and 5, we get 

➡️c/sinC=a/sinA —6 

Now,from equation 3 and 6, we get 

➡️   a/sinA=b/sinB=c/sinC 

🖕 

This is Sine Rule 

Cosine Rule 

By using this rule we can also find the sides of a triangle if the triangle is not right angled. 

So,let's start the derivation 

In ∆ABD 

AD²+x²=c² 

AD²=c²-x² —1 

In ∆ACD 

In ∆ACD 

AD²+(a-x)²=b² 

AD²=b²-(a-x)² —2 

Now, from equation 1 and 2, we get 

c²-x²=b²-a²-x²+2ax 

2ax=c²+a²-b² 

2acCosB=a²+c²-b²     (cosB=X/c then,X=cCosB)
  

➡️CosB=a²+c²-b²/2ac 

🖕 

This is cosine Rule 

We can find with any angle in a triangle

like,

CosA=b²+c²-a²/2bc,and 

CosC=a²+b²-c²/2ab 

  

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