Signs of the Trigonometric functions

Trigonometric functions of sine, cosine, tangent, cotangent are based on the signs of y and x coordinates in the respective four quadrants.

θ lies in which Quadrant I II III IV
Trigonometric functions
Sin θ +ve +ve -ve -ve
Cos θ +ve -ve -ve +ve
Tan θ +ve -ve +ve -ve
Cot θ +ve -ve +ve -ve
Cosec θ +ve +ve -ve -ve
Sec θ +ve -ve -ve +ve


From the above,

Quadrant I: The values of the trigonometric functions including Sine, Cosine, Tangent, Cotangent of any of the arc from the I Quadrant are the positive ones as coordinates are positive of the given points – P, S₁, and S₂, which define their specific values.




Quadrant II: From the 2nd quadrant points, for the arcs, P & S₂, both consist of negative abscissas (in the figure above), so, cotangent and the cosine exist as negative. Terminal point P’s ordinate is +ve so that Sine is +ve while ordinate of Point S₁ is -ve and hence, the tangent is negative.



Quadrant III: As abscissas plus terminal points P’s ordinates of arcs from the 3rd quadrant (see the figure above) are -ve. It follows the sine and cosine functions of the given arcs are -ve. Point S1’s ordinates plus point S2’s abscissa which belongs to arc from the III quadrant are +ve. Therefore, cotangent and tangent of the given arcs are +ve.

Quadrant IV: The functions including tangent, sine, and cotangent of the given arcs from IVth quadrant are -ve, similar to points S₁, S₂, and P’s coordinates which do belongs to them only. Only arc’s cosine functions from IVth quadrant is +ve are point P’s abscissas which belongs to them as you can see in the figure above.

Quadrant Values of arc Sin x Cos x Tan x Cot x
I From 0 to 90º + + + +
II From 90º to 180º +
III From 180º to 270º + +
IV From 270º to 360º +


Question 1: Give signs of the trigonometric functions

Solution: Sine 146º = Sin (90º + 56º) = As Sine lies from 0º to 90º, therefore, The sign will be positive.

Question 2: Cos 455º

Solution: Cos (360º + 95º) = Cos (90º + 5º) = Cos 5º

From the above solution, it clearly means that the sign is positive.

Question 3: Tan 573º

Solution: Tan (360º + 213º) = Tan (180º + 33º) = Tan 33º

Hence, Tan 573º is positive



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