Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/12) | Q#4

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Question

Angle x is such that sin x = a + b and cos x = a b, where a and b are constants.

A curve is such that . The point P (2,9) lies on the curve.

    i.       Show that a2 +b2 has a constant value for all values of x.

   ii.       In the case where tan x = 2, express a in terms of b.

Solution


i.
 

We are given that;

Taking squares of both sides of both equations;

We have the algebraic formula;

Adding both equations;

We have the trigonometric identity;

Therefore;

Hence, has a constant value for all values of x.

 

   ii.
 

We are given that;

Dividing both equations;

Since, ;

We are given that;

Hence;

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