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High School Mathematics - 2
10.8 Signs of Trigonometric Functions for Negative Angles

θsin θ cos θtan θ cosec θ sec θ cot θ
0010n.d.1n.d.
301/23/21/322/33
451/21/21221
603/21/232/321/3
9010n.d.1n.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
III IIIIV
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.
8/17
8/15
15/7

Q 2: Given sec x = 8/5 and 3π/2 < x < 2π, evaluate cot x.
15/8
8/5
5/8

Q 3: If cos x = 5/13 and x lies in the fourth quadrant, find the value of (2 - 3 cot x)/(4 - 9(sec2-1)1/2
-65/352
65/352
75/352

Q 4: State the quadrants in which Sin x = .5732 may lie.
II, III
I, II
I, IV

Q 5: State the quadrants in which tan x is negative and cos x is positive.
IV
II
III

Q 6: If A lies in the second quadrant and tan x = -5/12, find the value of 2 cos x/(1-sin x)
3
3/2
-3

Q 7: Given sec x = 8/5 and 3π/2 < x < 2π, evaluate cosec x.
- 8/(39)1/2
8/39
5/8

Q 8: State the quadrant in which cot x = 0.8736 may lie.
I, II
I, III
I, IV

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Question 10: This question is available to subscribers only!


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