Trigonometric ratios of any angle:

Consider system of rectangular co ordinate axes dividing plane into 4 quadrants. A line OP makes angle q with positive x-axis. The angle  q is said to be positive if measured in anticlockwise direction from positive x-axis and is negative if measured in the clockwise direction. The positive values of trigonometric ratios in various quadrants are shown; the signs of other ratios can be derived. Note that ∠xoy = Π/2, ∠xox' = Π , ∠xoy' = 3Π/2

P_iQ_i is positive if above x-axis, negative if below x-axis, OP_i is always taken positive. OQ_i is positive if along the positive x axis, negative if in opposite direction.

Therefore depending on signs of OQ_i and P_iQ_i the various trigonometrical ratios will have different signs.

Note:             

• The angle θ and 90°- θ  are complementary angles,  θ  and 180°- θ  are supplementary angles
• sin(nΠ + (-1)ⁿθ) = sinθ, n ∈ I 
• cos(2nΠ  ± θ) = cosθ, n ∈ I
• tan(nΠ + θ) = tanθ , n ∈ I

that is sine of general angle of form nΠ + (-1)ⁿθ will have same sign as that of sine of angle θ and so on. The same is true for respective reciprocal functions also.

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