If x is an acute angle and tan x =3/4 evaluate cos x - sin x/cos x + sin x
To find the value of the expression, let's first find the values of sin(x) and cos(x).
Given that tan(x) = 3/4, we can draw a right triangle with the opposite side measuring 3 and the adjacent side measuring 4. Using the Pythagorean theorem, we can find the length of the hypotenuse.
Hypotenuse² = Opposite² + Adjacent²
Hypotenuse² = 3² + 4²
Hypotenuse² = 9 + 16
Hypotenuse² = 25
Taking the square root of both sides, we have:
Hypotenuse = √25
Hypotenuse = 5
Since sin(x) = Opposite/Hypotenuse, we have:
sin(x) = 3/5
Similarly, cos(x) = Adjacent/Hypotenuse:
cos(x) = 4/5
Now, substitute the values of sin(x) and cos(x) into the expression:
(cos(x) - sin(x))/(cos(x) + sin(x))
= (4/5 - 3/5)/(4/5 + 3/5)
= (1/5)/(7/5)
= (1/5) * (5/7)
= 1/7
Therefore, cos(x) - sin(x)/cos(x) + sin(x) = 1/7.
To evaluate the expression cos x - sin x / cos x + sin x when tan x = 3/4, we can use the fact that tan x = sin x / cos x.
Let's start by finding the values of sin x and cos x. Since tan x = 3/4, we can set up the equation:
tan x = sin x / cos x
3/4 = sin x / cos x
To solve this equation, we can use the property of the tangent function:
tan x = sin x / cos x = (sin x) / (sqrt(1 - sin^2 x))
Plugging in the value of tan x, we have:
3/4 = (sin x) / (sqrt(1 - sin^2 x))
Cross-multiplying and simplifying, we get:
3 * sqrt(1 - sin^2 x) = 4 * sin x
Square both sides of the equation:
9 * (1 - sin^2 x) = 16 * sin^2 x
Expanding, we have:
9 - 9 * sin^2 x = 16 * sin^2 x
Simplifying further, we get:
25 * sin^2 x = 9
Divide both sides by 25:
sin^2 x = 9/25
Taking the square root of both sides:
sin x = sqrt(9/25) = 3/5
Now that we have the value of sin x, we can find cos x using the identity:
sin^2 x + cos^2 x = 1
Substituting the value of sin x, we have:
(3/5)^2 + cos^2 x = 1
9/25 + cos^2 x = 1
cos^2 x = 1 - 9/25
cos^2 x = 16/25
Taking the square root of both sides:
cos x = sqrt(16/25) = 4/5
Now we can substitute the values of sin x and cos x back into the original expression:
cos x - sin x / cos x + sin x
(4/5) - (3/5) / (4/5) + (3/5)
Simplifying each term:
(4 - 3) / 5 / (4 + 3) / 5
1/5 / 7/5
Invert the divisor and multiply:
1/5 * 5/7
1/7
Therefore, cos x - sin x / cos x + sin x = 1/7 when tan x = 3/4.