Question

find a square + b square​

Answer 1

Question:

Find the value of a^2 + b^2 if ;

a•sin@ - b•cos@ = 3 and

a•cos@ + b•sin@ = 4 .

Answer:

a^2 + b^2 = 25

Note:

• (sin@)^2 + (cos@)^2 = 1

• (tan@)^2 + 1 = (sec@)^2

• (cot@)^2 + 1 = (cosec@)^2

Solution:

We have,

a•sin@ - b•cos@ = 3 ---------(1)

a•cos@ + b•sin@ = 4 -----------(2)

Now,

Squaring both sides of eq-(1) , we get;

=> (a•sin@ - b•cos@)^2 = (3)^2

=> (a^2)•(sin@)^2 + (b^2)•(cos@)^2

– 2ab•sin@•cos@ = 9 --------(3)

Also,

Squaring both sides of eq-(2) , we get;

=> (a•cos@ + b•sin@)^2 = (4)^2

=> (a^2)•(cos@)^2 + (b^2)•(sin@)^2

+ 2ab•sin@•cos@ = 16 --------(4)

Now,

Adding eq-(3) and eq-(4), we get;

=> (a^2)•(sin@)^2 + (b^2)•(cos@)^2

– 2ab•sin@•cos@ + (a^2)•(cos@)^2

+ (b^2)•(sin@)^2 + 2ab•sin@•cos@

= 9 + 16

=> (a^2)•{(sin@)^2 + (cos@)^2}

+ (b^2)•{(sin@)^2 + (cos@)^2} = 25

=> a^2 + b^2 = 25.

{(sin@)^2 + (cos@)^2 = 1}

Hence,

The required value of a^2 + b^2 = 25 .