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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