Problema Solution
find sin theta and cos theta if the terminal side of theta lies along the line y=2xs in Quadrant 3?
Draw the angle 225 degrees in standard position, and then locate a convenient point on the terminal side of the angle. Use the coordinates of this point to find the sine, cosine, and tangent of the angle.
find the distance from the origin to the center of the circle. x^2+y^2-8x+6y=144.
find the remaining trigonometric functions of theta if csc theta=13/5 and cos theta
Answer provided by our tutors
- Find sin theta and cos theta if the terminal side of theta lies along the line y=2x in Quadrant 3?
let alfa = angle with initial side x = 0 and terminal side y = 2x thus tan alfa = 2 follows alfa = arctan(2)
theta = 180 + alfa
sin theta = sin (180 + alfa) = - sin alfa = - sin (arctan(2)) = - sin (63.43) = - 0.8944
cos theta = cos (180 + alfa) = - cos alfa = - cos (arctan(2)) = - cos (63.43) = - 0.4473
- Find the distance from the origin to the center of the circle x^2+y^2-8x+6y=144.
click here to see the graph of the circle
the equation of the circle can be written as
(x - 4)^2 + (y + 3)^2 = 13^2
the center of the circle is (4, -3)
we need to find the distance d from (0, 0) to (4, -3)
d^2 = (4 - 0)^2 + (-3 - 0)^2
d^2 = 16 + 9
d^2 = 25
d = 5
the distance from the origin to the center of the circle is 5.
- Find the remaining trigonometric functions of theta if csc theta=13/5 and cos theta.
csc theta = 13/5 and since csc theta = 1/(sin theta) we have
1/(sin theta) = 13/5
sin theta = 5/13
(cos theta)^2 = (1 - (sin theta)^2) (we are using the formula (sin theta)^2 + (cos theta)^2 = 1)
(cos theta)^2 = (1 - (5/13)^2)
cos theta = 12/13
cos theta = 0.9231
tan theta = (sin theta)/(cos theta)
tan theta = (5/13)/(12/13)
tan theta = 5/12
tan theta = 0.4167
cot theta = 1/(tan theta)
cot theta = 1/(5/12)
cot theta = 12/5
cot theta = 2.4