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Refer to the image and read this.
You will find the following equation in the steps.
(a²-b²)tan²(theta) -2actan(theta) +(c²-b²) = 0
If I assume tan(theta) = x, then the equations become,
(a²-b²)x² -2acx +(c²-b²) = 0
x² -2acx/(a²-b²) +(c²-b²)/ (a²-b²)= 0
Now i will relate this with the standard quadratic equation having roots a and b.
This quadratic equation is (x-a)(x-b)= x²-(a+b)x +ab
Here you can see that the coefficient of x is the sum of roots and the constant is the product of roots.
Now, according to what we have obtained,
a = tan alpha
b = tan beta,
so, a + b = tan alpha + tan beta
and ab = tan alpha× tan beta.
Now you can just apply the trigonometric formula and calculate what is required.
I hope you understand the approach.
All the best!
You will find the following equation in the steps.
(a²-b²)tan²(theta) -2actan(theta) +(c²-b²) = 0
If I assume tan(theta) = x, then the equations become,
(a²-b²)x² -2acx +(c²-b²) = 0
x² -2acx/(a²-b²) +(c²-b²)/ (a²-b²)= 0
Now i will relate this with the standard quadratic equation having roots a and b.
This quadratic equation is (x-a)(x-b)= x²-(a+b)x +ab
Here you can see that the coefficient of x is the sum of roots and the constant is the product of roots.
Now, according to what we have obtained,
a = tan alpha
b = tan beta,
so, a + b = tan alpha + tan beta
and ab = tan alpha× tan beta.
Now you can just apply the trigonometric formula and calculate what is required.
I hope you understand the approach.
All the best!
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john44:
thanks alot
please can you tell me how u obtained a= tanalpha and b= tan beta
for a trigonometric equation the roots are always the angles which.make the entire equation zero. similarly when i replaced tantheta with x, a and b became the roots of the equation.
since alpha and beta were the roots of trigonometric equation, tan alpha and tan beta become the roots of tye algebraic equation when tan theta is replaced by x
thanks alot
no problem
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