**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 + 16cos ^{2}θ =
9sin^{2}θ*

*or, 16cos ^{2}θ – 40cosθ + 25 = 9(1 –
cos^{2}θ)*

*or, 16cos ^{2}θ – 40cosθ + 25 = 9 – 9cos^{2}θ*

*or, 25cos ^{2}θ – 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 tan ^{2}A + cot^{2}A = 7.*

*Solution:** Here,*

* tanA + cotA = 3*

* Squaring both sides, we get*

* tan ^{2}A + 2tanA.cotA + cot^{2}A
= 9*

*or, tan ^{2}A + 2×1 + cot^{2}A =
9 [*

*∵*

*tanA × cotA = 1]**or, tan ^{2}A + 2 + cot^{2}A = 9*

*or, tan ^{2}A + cot^{2}A = 9 – 2
*

*or, tan ^{2}A + cot^{2}A =
7. Proved.*

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

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