In what interval(s) between 0 and 2π are tanθ and cosθ both positive?

Question

Please help

Answer 1

You really need to become very familiar with the unit circle (as in memorize). We draw the unit circle as a circle with radius "1" drawn with the center at the origin. Any point on the unit circle then has coordinates (cos(θ), sin(θ)) meaning that cos(θ) is the "x" and sin(θ) is the "y". So:

In Quadrant I, both sine and cosine are positive. In Quadrant II, cosine is negative but sine is positive. In Quadrant III, both sine and cosine are negative. In Quadrant IV, cosine is positive but sine is negative.

Since tan(θ) = sin(θ)/cos(θ) then tangent is positive in Q I (+/+ = +) and Q III (- / - = +) but negative in Q II (+/- = -) and Q IV (- / + = -)