I need help on both of these...
Find the exact value of cotx if the terminal side of angle x contains the point (-5,-12).
Find the exact value of secx for angle x that has a terminal side containing the point (-1,squareroot3)
First:
So the angle is in quadrant III
tan x = -12/-5 = 12/5, so
cot x = 5/12
2nd:
point is (-1,√3) which is in quad II
you have a right-angled triangle with sides 1, √3, and 2 ( the famous 30-60-90 triangle)
cos x = -1/2
sec x = -2
To find the exact value of cot(x) if the terminal side of angle x contains the point (-5, -12), we can use the trigonometric identity:
cot(x) = 1/tan(x)
First, we need to find the value of tan(x). We can use the tangent formula:
tan(x) = y/x,
where y is the y-coordinate (-12) and x is the x-coordinate (-5).
Plugging in the values:
tan(x) = -12/-5 = 12/5
Now, we can find the value of cot(x) by taking the reciprocal of tan(x):
cot(x) = 1/(12/5) = 5/12
Therefore, the exact value of cot(x) is 5/12.
To find the exact value of sec(x) for angle x that has a terminal side containing the point (-1, √3), we can use the trigonometric identity:
sec(x) = 1/cos(x)
First, we need to find the value of cos(x). We can use the cosine formula:
cos(x) = x/r,
where x is the x-coordinate (-1) and r is the distance from the origin to the point (-1, √3). To find r, we can use the Pythagorean theorem:
r = √(x^2 + y^2) = √((-1)^2 + (√3)^2) = √(1 + 3) = √4 = 2
Now, we can find the value of cos(x) by dividing x by r:
cos(x) = -1/2
Finally, we can find the value of sec(x) by taking the reciprocal of cos(x):
sec(x) = 1/(-1/2) = -2
Therefore, the exact value of sec(x) is -2.