# 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;