Trig Functions in Four Quadrants

The angle θ is an acute angle (θ < 90°) and is measured with reference to the positive x-axis, in the anticlockwise direction. Further, these trig functions have different numeric signs (+ or -) in the different quadrants, which are based on the positive or negative axis of the quadrant. The trigonometric functions of Sinθ, Cosecθ are positive in quadrants I and II, and are negative in quadrants III and IV. All the trigonometric functions have a positive range in the first quadrant. The trigonometric functions Tanθ, Cotθ are positive only in Quadrants I and III, and the trigonometric ratios of Cosθ, Secθ are positive only in quadrants I and IV.

The trigonometric functions have values of θ, (90° – θ) in the first quadrant. The cofunction identities provide the interrelationship between the different complementary trigonometric functions for the angle (90° – θ).

• sin(90°−θ) = cos θ
• cos(90°−θ) = sin θ
• tan(90°−θ) = cot θ
• cot(90°−θ) = tan θ
• sec(90°−θ) = cosec θ
• cosec(90°−θ) = sec θ

The domain θ value for different trigonometric function in the second quadrant is (π/2 + θ, π – θ), in the third quadrant is (π + θ, 3π/2 – θ), and in the fourth quadrant is (3π/2 + θ, 2π – θ). For π/2, 3π/2 the trigonometric values changes as their complementary ratios such as Sinθ⇔Cosθ, Tanθ⇔Cotθ, Secθ⇔Cosecθ. For π, 2π the trigonometric values remain the same. The changing trigonometric ratios in different quadrants and angles can be understood from the below table.