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### High School Mathematics - 210.8 Signs of Trigonometric Functions for Negative Angles

 θ sin θ cos θ tan θ cosec θ sec θ cot θ 0° 0 1 0 n.d. 1 n.d. 30° 1/2 Ö3/2 1/Ö3 2 2/Ö3 Ö3 45° 1/Ö2 1/Ö2 1 Ö2 Ö2 1 60° Ö3/2 1/2 Ö3 2/Ö3 2 1/Ö3 90° 1 0 n.d. 1 n.d. 0
Signs of trigonometric functions for negative angles
Let P(x,y) be a point on the unit circle with centre at origin such that angle AOP = θ. If angle AOQ = -θ, then the co-ordinates of the point Q will be (x,-y). Hence
cos(-θ) = x = cosθ
sin(-θ) = -y = -sinθ
 I II III IV sinθ + + - - cosθ + - - + tanθ + - + - cosecθ + + - - secθ + - - + cotθ + - + -
In the first quadrant, as θ increases from 0 to π/2, sinθ increases from 0 to 1.
In the second quadrant, as θ increases from to π/2 to π,to π sinθ decreases from 1 to 0.
In the third quadrant, as θ increases from π to 3π/2, sinθ decreases from 0 to -1.
In the fourth quadrant, sinθ increases from -1 to 0, as θ increases from 3π/2 to 2π.

Directions: Solve the following.
 Q 1: Given that sin x = 8/17 and x lies in quadrant I, find the value of tan x.15/78/178/15 Q 2: Given sec x = 8/5 and 3π/2 < x < 2π, evaluate cosec x.5/8- 8/(39)1/28/39 Q 3: State the quadrants in which tan x is negative and cos x is positive.IIIIVII Q 4: State the quadrant in which cot x = 0.8736 may lie.I, IVI, IIII, II Q 5: If cos x = 5/13 and x lies in the fourth quadrant, find the value of (2 - 3 cot x)/(4 - 9(sec2-1)1/275/35265/352-65/352 Q 6: Given sec x = 8/5 and 3π/2 < x < 2π, evaluate cot x.8/55/815/8 Q 7: If A lies in the second quadrant and tan x = -5/12, find the value of 2 cos x/(1-sin x)3/2-33 Q 8: State the quadrant in which cosec x = 4.1477 may lie.II, IIII, IVI, II Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!

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