If tan(x)=8/3 and sin(x)<0, then find cos (x) sec(x) csc(x) i don't understand how u do this problem
Think of the various definitions of these functions in terms of the height, base and hypoteneuse of a right-angled triangle. You're told that tan(x)=8/3, which means that the ratio of the height to the side is 8 to 3. So you can work out the hypoteneuse by Pythagoras. That should enable you to work out the other functions. Don't forget that condition about the sine of x being less than 0 though: there's more than one angle in a 360-degree circle with a tangent of 8/3, and you need to make sure you pick the right one.
If tan x is positive and sin x is negative then yx must be in the third quadrant. For tan x = 8/3, the reference angle to the -x axis must be arcsin 8/sqrt(8^2 + 3^2) = 69.44 degrees. That means the angle x = 249.44 degrees. The cosine of that angle is -0.3512. The secant is the reciprocal of cosine. csc is the reciprocal of the sin, or -(sqrt73)/8
To find the values of cos(x), sec(x), and csc(x) given that tan(x) = 8/3 and sin(x) < 0, we need to use the trigonometric identities.
First, let's analyze tan(x) = 8/3. Tangent is defined as the ratio of sine to cosine, so we can write:
tan(x) = sin(x) / cos(x)
Since sin(x) < 0 and tan(x) = 8/3, we know that sin(x) is negative in the quadrant where the tangent value is positive. This means that x is in the second quadrant.
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can find the value of cos(x) by substituting sin(x) = -√(1 - cos^2(x)):
(8/3)^2 = (-√(1 - cos^2(x))) / cos(x)^2
Simplifying this equation, we get:
64/9 = (1 - cos^2(x)) / cos^2(x)
Multiplying through by cos^2(x), we have:
64 cos^2(x) = 9 - 9 cos^2(x)
Bringing all terms to one side, we have a quadratic equation:
73 cos^2(x) - 9 = 0
Solving this quadratic equation will give us the value(s) for cos(x) in the second quadrant. By factoring or using the quadratic formula, we find:
cos(x) = ± √(9/73)
Now that we have the value of cos(x), we can find sec(x) and csc(x) using the reciprocal relationships:
sec(x) = 1/cos(x)
csc(x) = 1/sin(x)
So, substitute cos(x) = √(9/73) and sin(x) = -√(1 - cos^2(x)) into the equations for sec(x) and csc(x) to get the values.