Trigonometric ratios of 270 degrees minus theta
(270 -  θ)  

In the previous section we have already learnt Trigonometric ratios of (270 + θ)
Now here we will discuss about Trigonometric ratios of 270 degrees minus theta (270 - θ)
In trigonometric ratios, we will find the relation between six trigonometric ratios.
sin( 90 - $\Theta) = cos \Theta$
cos( 90 - $\Theta) = sin \Theta$
tan( 90 -$\Theta) = cot \Theta$
csc( 90 - $\Theta) = sec \Theta$
sec( 90 - $\Theta) = csc \Theta$
cot( 90 - $\Theta) = tan \Theta$

sin( 180 - $\Theta) = sin \Theta$
cos( 180 - $\Theta) = -cos \Theta$
tan( 180 -$\Theta) = -tan \Theta$
csc( 180 - $\Theta) = csc\Theta$
sec( 180 - $\Theta) = -sec\Theta$
cot( 180- $\Theta) = -cot \Theta$

sin( 90 + $\Theta) = cos \Theta$
cos( 90 + $\Theta) = -sin \Theta$
tan( 90 + $\Theta) = -cot \Theta$
csc( 90 + $\Theta) = sec \Theta$
sec( 90 + $\Theta) = -csc \Theta$
cot( 90 + $\Theta) = -tan \Theta$

sin( 180 + $\Theta) = -sin \Theta$
cos( 180 + $\Theta) = -cos \Theta$
tan( 180 +$\Theta) = tan \Theta$
csc( 180 + $\Theta) = -csc\Theta$
sec( 180 + $\Theta) = -sec\Theta$
cot( 180+ $\Theta) = cot \Theta$

Using the above results we can easily find out the trigonometric ratios of 270 degrees plus theta (270 - $\Theta$)
1) sin( $270 - \Theta ) = sin [ 180 + ( 90 - \Theta$)]
But we know that sin(180 + $\Theta)= - sin \Theta$)
∴ sin [ 180 + ( 90 - $\Theta)] = -sin (90 -\Theta$)
But sin (90 -$\Theta) = cos \Theta$)
∴ $\underline {sin( 270 - \Theta )= - cos \Theta}$

2) cos( $270 - \Theta ) = cos [ 180 + ( 90 - \Theta$)]
But we know that cos(180 + $\Theta)= - cos \Theta$)
∴ cos [ 180 + ( 90 - $\Theta)] = -cos (90 -\Theta$)
But cos (90 -$\Theta) = sin \Theta$)
∴ $\underline {cos( 270 - \Theta )= -sin \Theta}$

3) tan( $270 - \Theta ) = tan [ 180 + ( 90 - \Theta$)]
But we know that tan(180 + $\Theta)= tan \Theta$)
∴ tan[ 180 + ( 90 -$\Theta)] = tan (90 -\Theta$)
But tan (90 -$\Theta) = cot \Theta$)
∴ $\underline {tan( 270 - \Theta )= cot \Theta}$

4) $csc( 270 - \Theta ) = \frac{1}{sin( 270 + \Theta )}$
But $sin( 270 - \Theta )= - cos \Theta$
∴ csc( $270- \Theta )= \frac{1}{-cos \Theta}$
∴ $\underline {csc( 270 - \Theta )= - sec \Theta}$

5) $sec( 270 - \Theta ) = \frac{1}{cos( 270 + \Theta )}$
But $cos( 270 - \Theta )= -sin \Theta$
∴ sec( $270 - \Theta )= -\frac{1}{sin \Theta}$
∴ $\underline {sec( 270 - \Theta )=-csc \Theta}$

6) $cot( 270 - \Theta ) = \frac{1}{sin( 270 - \Theta )}$
But $tan( 270 - \Theta )= cot \Theta$
∴ csc( $270 - \Theta )= \frac{1}{-cos \Theta}$
∴ $\underline {cot( 270 - \Theta )= tan \Theta}$

Examples on Trigonometric ratios of 270 degrees minus theta
(270 - θ) 

1) Find the value of $cos 270^{0}$
Solution : $cos 270^{0}$
270 = 270 - 0
As we know that $cos (270 - \Theta) = -sin\Theta$
∴ $cos 270^{0}$ = cos (270 -0) = - sin0
⇒ $cos 270^{0}$ = 0

2) Find the value of $csc 225^{0}$
Solution : $csc 225^{0}$
225 = 270 - 45
As we know that $csc (270 - \Theta) = -sec\Theta$
∴ $csc 225^{0}$ = csc (270 -45) = - sec 45
⇒ $csc 225 = -{\sqrt{2}}$


11th grade math

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