Trigonometric functions in different Quadrants
The trigonometric functions in different quadrants, for that we will use a unit circle. The circle whose radius is 1 is called unit circle in trigonometry. The center of this circle is origin(0,0). As the radius is 1 we can directly measure sine,cosine and tangent.
Let X'OX and YOY' be the coordinate axes. Draw a circle with radius 1. Suppose the circle cut the axes at A,B,A' and B'.Let P(x,y) be any point on the circle such that $\angle AOP = \Theta$.
As OP = 1, radius of the unit circle.
⇒ $cos \Theta = \frac{adjacent}{hypotenuse} = \frac{x}{OP} =\frac{x}{1}= x$
and $sin \Theta = \frac{opposite}{hypotenuse} = \frac{y}{OP} =\frac{y}{1}= y$
From the diagram its clear that,
1 $\leq $ x $\leq $ 1 and 1 $\leq $ y $\leq $ 1
OR 1 $\leq cos \Theta \leq $ 1 and 1 $\leq sin \Theta \leq $ 1 for all values of $\Theta$.
In the first quadrant as the angle $\Theta$ increases from $0^{0}$ to $90^{0}$ so in this quadrant 'y' values increases from 0 to 1 so $sin \Theta$ increases from 0 to 1. In the second quadrant as the angle $\Theta$ increases from $90^{0}$ to $180^{0}$ so in this quadrant 'y' values decreases from 1 to 0 so $sin \Theta$ decreases from 1 to 0. In the third quadrant as the angle $\Theta$ increases from $180^{0}$ to $270^{0}$ so $sin \Theta$ decreases from 0 to 1. In the fourth quadrant as the angle $\Theta$ increases from $270^{0}$ to $360^{0}$ so $sin \Theta$ increases from 1 to 0.
Values of trigonometric functions in different quadrants

I QUADRANT 

II QUADRANT 
$sin \Theta$ 
increases from 0 to 1 
$sin \Theta$ 
decreases from 1 to 0 
$cos \Theta$ 
decreases from 1 to 0 
$cos \Theta$ 
decreases from 0 to 1 
$tan \Theta$ 
increases from 0 to $\infty$ 
$tan \Theta$ 
increases from $\infty$ to 0 
$cot \Theta$ 
decreases from $\infty$ to 0 
$cot \Theta$ 
decreases from 0 to $\infty$ 
$sec\Theta$ 
increases from 1 to $\infty$ 
$sec\Theta$ 
increases from $\infty$ to 1 
$csc\Theta$ 
decreases from $\infty$ to 1 
$csc\Theta$ 
decreases from 1 to $\infty$ 

III QUADRANT 

IV QUADRANT 
$sin \Theta$ 
increases from 0 to 1 
$sin \Theta$ 
increases from 1 to 0 
$cos \Theta$ 
decreases from 1 to 0 
$cos \Theta$ 
increases from 0 to 1 
$tan \Theta$ 
increases from 0 to $\infty$ 
$tan \Theta$ 
increases from $\infty$ to 0 
$cot \Theta$ 
decreases from $\infty$ to 0 
$cot \Theta$ 
decreases from 0 to $\infty$ 
$sec\Theta$ 
decreases from 1 to $\infty$ 
$sec\Theta$ 
dencreases from $\infty$ to 1 
$csc\Theta$ 
decreases from $\infty$ to 1 
$csc\Theta$ 
decreases from 1 to $\infty$ 
Note : $\infty$ and $\infty$ are not real numbers. When we say that $tan \Theta$ increases from 0 to $\infty$ for as $\Theta$ varies from 0 to $\pi$/2 that means $tan \Theta$ increases in the interval (0,$\pi$/2 ) and attains arbitrarily large positive values of $\Theta$ tends to $\pi$/2. similarly for other trigonometric functions.
11th grade math
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