Question

It is given tan 50=k

Write cos 40 in terms of k

Also do the Cartesian plane if possible

Answer 1

tan 50° = k. Since cos 40° = sin 50°, write sin 50° in terms of tan 50°:

1 + tan^2 50° = sec^2 50° = 1 / cos^2 50°,
so cos 50° = 1 / sqrt(1 + k^2) (positive because 50° is in QI),
and sin 50° = tan 50° · cos 50° = k / sqrt(1 + k^2).

Therefore
cos 40° = sin 50° = k / sqrt(1 + k^2).

Cartesian-plane/geometric view:
Take the ray making 50° with the positive x-axis and a point on it with coordinates proportional to (1, k) because tan = y/x = k. Normalizing by the hypotenuse sqrt(1 + k^2) gives the unit-circle coordinates (cos 50°, sin 50°) = (1/√(1+k^2), k/√(1+k^2)), so cos 40° = sin 50° = k/√(1+k^2).