Trigonometry : Practical application.
If x is an acute angle, and tan x = 3\4, evaluate.
cos x - sin x \cos x + sin x
I don't understand
if tan x = 3/4 then in a right triangle opposite x is 3 and adjacent is 4
then draw a right triangle with sides 3 and 4 and hypotenuse = sqrt(9+16) = 5
3, 4 , 5 right triangle
sin x = 3/5
cos x = 4/5
Now do it.
To evaluate the expression cos x - sin x / cos x + sin x, we need to use the trigonometric identity called the Pythagorean identity.
The Pythagorean identity is given by: cos^2(x) + sin^2(x) = 1
To find cos x and sin x, we can use the given information. tan x = 3/4 means that the ratio of sin x / cos x is 3/4.
We can represent this information in the following way:
sin x / cos x = 3/4
Now, we can square both sides of the equation to get rid of the fractions:
(sin x / cos x)^2 = (3/4)^2
(sinx)^2 / (cosx)^2 = 9/16
Using the Pythagorean identity, we know that (cos x)^2 = 1 - (sin x)^2. So we can substitute this expression into our equation:
(sinx)^2 / (1 - (sinx)^2) = 9/16
Now, we can cross-multiply to solve for (sin x)^2:
16(sin x)^2 = 9(1 - (sin x)^2)
Distributing the 9:
16(sin x)^2 = 9 - 9(sin x)^2
Add 9(sin x)^2 to both sides:
25(sin x)^2 = 9
Divide by 25:
(sin x)^2 = 9/25
Taking the square root of both sides:
sin x = ±√(9/25)
Simplifying:
sin x = ±3/5
Since x is an acute angle, sine x is positive in the first and second quadrants. So, we take the positive value:
sin x = 3/5
Now, we can find cos x using the equation sin^2(x) + cos^2(x) = 1:
(3/5)^2 + cos^2(x) = 1
9/25 + cos^2(x) = 1
Subtract 9/25 from both sides:
cos^2(x) = 1 - 9/25
cos^2(x) = 16/25
Taking the square root of both sides:
cos x = ±√(16/25)
Simplifying:
cos x = ±4/5
Since x is an acute angle, cosine x is positive in the first quadrant. So, we take the positive value:
cos x = 4/5
Now, we can substitute these values into the given 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, the value of the expression cos x - sin x / cos x + sin x, with the given conditions, is 1/7.