If tan x =5/12,0 degrees < x < 90 degrees.evaluate,without using mathematical tables or calculator, sin X/(sin X)^2 + cos x
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.
To evaluate sin x/(sin x)^2 + cos x without using a calculator or mathematical tables, we will use the given information that tan x = 5/12.
We can start by finding the values of sin x and cos x using the given value of tan x. Recall the trigonometric identity: tan x = sin x / cos x.
We know tan x = 5/12, so we can substitute this value into the identity:
5/12 = sin x / cos x
To eliminate the fractions, we can multiply both sides by 12 cos x:
12 cos x * (5/12) = sin x
The 12's cancel out, leaving us with:
5 cos x = sin x
Now we have an equation relating sin x and cos x. We can square both sides to eliminate the sin x term:
(5 cos x)^2 = (sin x)^2
25 cos^2 x = sin^2 x
Now, let's substitute these values into the original expression sin x/(sin x)^2 + cos x:
sin x / (sin x)^2 + cos x
Using the relationship we found earlier, we can substitute in sin^2 x for (25 cos^2 x):
sin x / (25 cos^2 x) + cos x
Since the expression is already simplified, this is our final answer:
sin x / (25 cos^2 x) + cos x