Question 946102
tan(q) = -.09
q is in quadrant 4.
find the value of cos(q).


if q was in quadrant 1, it's tangent would be .09


using your calculator, the angle in quadrant 1 whose tangent is .09 is equal to 5.142764558 degrees.


5.142764558 degrees is the reference angle.


the equivalent angle in quadrant 4 is equal to 360 - 5.142764558 which is equal to 354.8572354 degrees.


using your calculator, the cosine of 354.8572354 degrees is equal to .9959744388.


you can confirm using your calculator.


tan(354.8572354) = -.09
cos(354.8572354) = .9959744388


there is another way to find it without resorting to using the calculator.


this involves knowledge of the unit circle.


in the unit circle, the hypotenuse of a triangle is always equal to 1.


you know that tan(q) is equal to -.09


the tangent function in the unit circle is equal to y/x.


you get y/x = -.09
solve for y to get y = -.09x.


your x value in the unit circle is equal to x.
your y value in the unit circle is equal to -.09x.
your hypotenuse in the unit circle is always equal to 1.


by pythagorus, x^2 + y^2 = h^2 where h is the hypotenuse of the right triangle formed by x and y.


this equation becomes x^2 + (-.09x)^2 = 1^2


simplify to get x^2 + .0081x^2 = 1


combine like terms to get 1.0081x^2 = 1


divide both sides of this equation by 1.0081 to get x^2 = 1/.0081 = .9919650828.


take the square root of both sides of this equation to get x = .9959744388.


solve for y to get y = -.09x = -.09 * .9959744388 = -.0896376995


you now have:
hypotenuse = 1
x = .9959744388
y = -.0896376995

tan(q) = y/x = -.0896376995 / .9959744388 = -,09


cos(q) = x / h = x / 1 = x = .9959744388.