# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/11) | Q#1

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Question

The acute angle  radians is such that tan , where  is a positive constant. Express, in terms of ,

i.

ii.

iii.

Solution

i.

From the diagram below it is evident that if  is acute angle i.e.  lies in the first quadrant, then  will be in second quadrant.

From the basic trigonometry it is known that  in second quadrant is negative i.e. .

Therefore;

Since

We can write;

ii.

We have the trigonometric relation;

We can rewrite it as;

The sine of angles is equal to cosine of its complement, and vice versa;

Therefore;

Hence

We have the trigonometric relation;

We are given that;

Therefore;

iii.

We have the trigonometric relation;

Therefore;

We have the trigonometric identity;

Dividing both sides by ;

We have the trigonometric relation;

Hence we can write;

We are given that;

Therefore;