let (5,-4) Be a point on the terminal side of zero find the exact values of cos
To find the exact value of cos, we need to determine the angle formed by the terminal side of zero and the line connecting the origin (0,0) to the given point (5,-4). To do this, we can use the concept of inverse trigonometric functions.
The given point (5,-4) lies in the fourth quadrant as the x-coordinate is positive and the y-coordinate is negative. We can find the angle by using the tangent ratio:
tan(theta) = (opposite/adjacent) = y/x = (-4/5)
Now, we need to find the angle whose tangent is (-4/5). We can use the inverse tangent function (also known as arctan or tan^(-1)) to calculate this:
theta = arctan(-4/5)
Using a scientific calculator or a trigonometric table, we can find the value of arctan(-4/5) to be approximately -0.67474094 radians (or -38.66 degrees).
Since cos(theta) is the x-coordinate of the given point divided by the length of the line connecting the origin to the point, we can calculate it as follows:
cos(theta) = x/r
Given that the x-coordinate is 5 and r is the distance from the origin to the point, which is the square root of the sum of the squares of the x and y-coordinates:
r = sqrt(x^2 + y^2) = sqrt(5^2 + (-4)^2) = sqrt(25 + 16) = sqrt(41)
Therefore:
cos(theta) = x/r = 5/sqrt(41)
Hence, the exact value of cos(theta), when the point (5,-4) is on the terminal side of zero, is 5/sqrt(41).
Draw the angle. You have
x = 5
y = -4
r = √41
cosθ = x/r = ____