## Conversion of Trigonometric Ratios

There are six trigonometric ratios. If one of them is known, the remaining five can be obtained by the conversion of trigonometric ratios. If we know any one of the trigonometric ratio, then the rest can be converted in terms of that known ratio. This can be done by using either of the following two methods:

(a) Using basic trigonometric relations

(b) Using Pythagoras theorem

### Lets study the following worked out examples:

Example 4: If 3Sinθ + 4cosθ = 5, show that cosθ = 4/5.

Solution: Here,

3sinθ + 4cosθ = 5

or,     5 – 4cosθ = 3sinθ

Squaring both sides, we get

25 – 40cosθ + 16cos2θ = 9sin2θ

or,     16cos2θ – 40cosθ + 25 = 9(1 – cos2θ)

or,     16cos2θ – 40cosθ + 25 = 9 – 9cos2θ

or,     25cos2θ – 40cosθ + 16 = 0

or,     (5cosθ – 4)2 = 0

or,     5cosθ – 4 = 0

or,     5cosθ = 4

or,     cosθ = 4/5. Shown.

Example 7: If tanA + cotA = 3, prove that tan2A + cot2A = 7.

Solution: Here,

tanA + cotA = 3

Squaring both sides, we get

tan2A + 2tanA.cotA + cot2A = 9

or,     tan2A + 2×1 + cot2A = 9   [ tanA × cotA = 1]

or,     tan2A + 2 + cot2A = 9

or,     tan2A + cot2A = 9 – 2

or,     tan2A + cot2A = 7.  Proved.

You can comment your questions or problems regarding the conversion of trigonometric ratios here.