What do you use the Tangent rule for?

The tangent rule shows the relationship among two sides a,b and two angles A,B

of a triangle ABC . It is (a-b)/(a+b) = tan((A-B)/2) / tan((A+B)/2) .

Think the case that we know two sides a,b and one angle C between them .

When we want to know A and B , we know A+B = 180°-C and (A+B)/2 = 90°-C/2 .

And by the tangent rule ,

tan((A-B)/2)

= [(a-b)/(a+b)]tan((A+B)/2)

= [(a-b)/(a+b)]tan(90°-C/2)

= [(a-b)/(a+b)][1/tan(C/2)]

So we can find tan((A-B)/2) , therefore we can find (A-B)/2 .

A = (A+B)/2 + (A-B)/2 and B = (A+B)/2 - (A-B)/2 , so we can find A and B .

But we can find A and B without the tangent rule .

Find c using the cosine rule c^2 = a^2 + b^2 - 2ab*cosC ,

and find A and B using the sine rule a/sinA = b/sinB = c/sinC .

There's a natural question, when you learn about the sine and cosine rules: "Is there a tan rule?"

The answer to that is yes - yes, there is a tan rule.

The natural follow-up is "Why don't we learn it?" Maybe you can memorize it!

you use it for math aka trigonometry