If tan x =5/12,0 degrees < x < 90 degrees.evaluate,without using mathematical tables or calculator, sin X/(sin X)^2 + cos x

Answer 1

Given that tan x = 5/12, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to find cos x.

Since tan x = 5/12, we can apply the definition of tangent which is sin x/cos x.

tan x = sin x/cos x,
5/12 = sin x/cos x.

Now, we need to find cos x. Squaring both sides of the equation and utilizing the Pythagorean identity, we get:

(5/12)^2 = (sin x/cos x)^2,
25/144 = (sin^2 x)/(cos^2 x),
25(cos^2 x) = 144(sin^2 x).

Using the identity sin^2 x + cos^2 x = 1, we can replace sin^2 x in the equation:

25(cos^2 x) = 144(1 - cos^2 x),
25 cos^2 x = 144 - 144(cos^2 x),
25 cos^2 x + 144 cos^2 x = 144,
169 cos^2 x = 144.

Dividing by 169, we have:

cos^2 x = 144/169.

Taking the square root of both sides, we find:

cos x = ±12/13.

Since x lies in the first quadrant (0 degrees < x < 90 degrees), cos x will only be positive. Therefore, cos x = 12/13.

Now, we can substitute sin x and cos x into the expression sin x/(sin x)^2 + cos x:

sin x/(sin x)^2 + cos x = sin x/(sin^2 x + cos x).

Since sin^2 x + cos^2 x = 1, we have:

sin x/(1 + cos x).

Substituting cos x = 12/13, we get:

sin x/(1 + 12/13) = sin x/(25/13) = (13/25) sin x.

Therefore, sin x/(sin x)^2 + cos x = (13/25) sin x.