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TRIGONOMETRIC RATIOS OF COMPLEMENTARY ANGLES

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Introduction to Complemtary Anglestrig_ratio_1

What are Complementary Angles


A pair of angles is complementary if the sum of their measures is 90 degrees.

In the adjoining Δ ABC, ∠ A and ∠ C are pair of complementary angles. Following this we have, ∠ A + ∠ C = 90°.


Also, ∠ C = 90° - ∠ A…(i)


We already are aware of trigonometric ratios, now we will define all six trigonometric ratios with respect to ∠ A and ∠ C.

trig_ratio_2



Trigonometric ratios with respect to ∠ A

Sin A = BC / AC
Cos A = AB / AC
Tan A = BC / AB         (I)
Cosec A = AC / BC 
Sec A = AC / AB
Cot A = AB / BC   



Trigonometric ratios with respect to ∠ C

Sin C = AB / AC
Cos C = BC / AC
Tan C = AB / BC         (II)
Cosec C = AC / AB
Sec C = AC / BC
Cot C = BC / AB

Substituting C = 90° – A (from (i))

Sin (90° – A) = AB / AC
Cos (90° – A) = BC / AC
Tan (90° – A) = AB / BC         (III)
Cosec (90° – A) = AC / AB
Sec (90° – A) = AC / BC
Cot (90° – A) = BC / AB

Now, compare the ratios in (I) and (III)


Sin (90° – A) = AB / AC = Cos A

Cos (90° – A) = BC / AC = Sin A
Tan (90° – A) = AB / BC = Cot A
Cosec (90° – A) = AC / AB = Sec A
Sec (90° – A) = AC / BC = Cosec A
Cot (90° – A) = BC / AB = Tan A

So,

Sin (90° – A) = Cos A
Cos (90° – A) = Sin A
Tan (90° – A) = Cot A
Cosec (90° – A) = Sec A
Sec (90° – A) = Cosec A
Cot (90° – A) = Tan A

For all values of angle A lying between 0° and 90°.


Now, we will check whether this holds for A = 0° or A = 90°


Tan 0° = 0 = Cot 90°

Sec 0° = 1 = Cosec 90°
Sec 90°, Cosec 90°, Tan 90° and Cot 90° are not defined.

On the basis of above discussion, we will solve the following problem:

Evaluate: Tan 65°
               Cot 25°

We know: Cot A = Tan (90° – A)

Cot 25° = Tan (90° - 25°) = Tan 65°

That is, Tan 65° = Tan 65° = 1

            Cot 65°     Tan 65°



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