If tanA = 1/2 then evaluate cos A/sinA +sinA /1+cosA

Mathematics

1 answer:

Deffense [45]3 years ago

3 0

Answer:

√5.

Step-by-step explanation:

Tan A = 1/2 means that the right triangle containing angle A has legs of length 1 and 2 units. So the hypotenuse = √(1^2 + 2^2) = √5 (using the Pythagoras theorem). The side opposite to < A = 1 unit and the adjacent side = 2 (as tan = opposite / adjacent).

so cos A = adjacent / hypotenuse = 2/√5.

and sin A = opposite / hypotenuse = 1 / √5

cos A / sin A = 2/√5 / 1/ √5 = 2.

sin A / (1 + cos A) = 1/√5 (1 + 2/ √5)

= 1 / √5 ( (√5 + 2) /√5)

= 1 / (√5 + 2)

So the answer is:

2 + 1 /(√5 + 2).

We can simplify it further by multiplying top and bottom of the fraction by the complement of √5 + 2 which is √5 - 2.

2 + 1 / (√5 + 2)

= 2(√5 + 2) + 1 / (√5 + 2 )

= { 2(√5 + 2) + 1 } / (√5 + 2)

Multiplying this by √5 - 2 / √5 - 2 we get:

(2(5 - 4) + √5 - 2) / (5 -4)

= 2 + √5 - 2 / 1

= √5.

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