Given that sinx= 3/5 and that x terminates in Quadrant 2 , determine the values for cosx and tanx
Find Cos x/2
sin A = 3/5 = y/r, Q2.
x^2 + y^2 = r^2
x^2 = r^2 - y^2 = 5^2 - 3^2=25 - 9 = 16
X = -4, Q2.
Cos A = X/r = -4/5
Tan A = Y/X = 3/-4 = -3/4
To find the values of cosx and tanx, we can use the given information that sinx = 3/5.
Since sinx = opposite/hypotenuse, we can draw a right triangle in Quadrant 2 where the opposite side is 3 and the hypotenuse is 5.
Using the Pythagorean theorem, we can find the adjacent side:
adjacent^2 + opposite^2 = hypotenuse^2
adjacent^2 + 3^2 = 5^2
adjacent^2 + 9 = 25
adjacent^2 = 25 - 9
adjacent^2 = 16
adjacent = √16
adjacent = 4
So, in Quadrant 2, the values are:
cosx = adjacent/hypotenuse = 4/5
tanx = opposite/adjacent = 3/4
To find cos(x/2), we can use the half-angle formula for cosine:
cos(x/2) = ±√((1 + cosx)/2)
Since x is in Quadrant 2, cosx is negative. So, we can use the negative sign in the formula.
cos(x/2) = -√((1 + cosx)/2)
cos(x/2) = -√((1 + 4/5)/2)
cos(x/2) = -√((9/5)/2)
cos(x/2) = -√(9/10)
cos(x/2) = -√9/√10
cos(x/2) = -3/√10
cos(x/2) = -3√10/10
Therefore, cos(x/2) = -3√10/10.
To find the values of cos(x) and tan(x), we can use the given information about sin(x) and the quadrant that x terminates in.
Given that sin(x) = 3/5 and x terminates in Quadrant 2, we know that sin(x) is positive in Quadrant 2, and sin(x) = y/r, where y is the y-coordinate and r is the hypotenuse. Therefore, we can determine the values of the remaining sides of the right triangle formed in Quadrant 2.
Let's denote the opposite side as y and the hypotenuse as r. Using the Pythagorean theorem, we can find the adjacent side:
cos(x) = sqrt(1 - sin^2(x))
cos(x) = sqrt(1 - (3/5)^2)
cos(x) = sqrt(1 - 9/25)
cos(x) = sqrt(25/25 - 9/25)
cos(x) = sqrt(16/25)
cos(x) = 4/5
So, cos(x) = 4/5.
Now, to find tan(x):
tan(x) = sin(x) / cos(x)
tan(x) = (3/5) / (4/5)
tan(x) = 3/5 * 5/4
tan(x) = 15/20
tan(x) = 3/4
So, tan(x) = 3/4.
Next, let's find cos(x/2). Using the half-angle identity formula for cosine:
cos(x/2) = sqrt((1 + cos(x)) / 2)
Substitute in the value of cos(x):
cos(x/2) = sqrt((1 + 4/5) / 2)
cos(x/2) = sqrt((9/5) / 2)
cos(x/2) = sqrt(9/10)
cos(x/2) = 3/√10
cos(x/2) = (3√10) / 10
Therefore, cos(x/2) = (3√10) / 10.