# csc a=2, pi/2<a<pi find sin a, cos a, tan a

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### 2 Answers

given: cosec a = 2 and `pi/2 < a < pi`

cosec is inverse of sine, thus

csc a = 1/sin a = 2

or, sin a = 1/2

cos a = `sqrt (1-sin^2 a) = sqrt(1-(1/2)^2) = sqrt (1-1/4) = sqrt (3/4) = - sqrt 3/2`

and tan a = `sina /cosa = (1/2)/(-sqrt3/2) = -1/sqrt3`

we chose negative signs for cos and tan, since a is between pi/2 and pi, and between these ranges, cos a and tan a will be negative.

Hope this helps.

From the below tables givne in the attachments we can easily find the values of **sin a, cos a, tan a**

**Given ,**

csc a= 2, `pi/2<a<pi` ie the **angle a** is in the **second quadrant .**

From the **table I , we get the values** and from the **table II , we assign the signs** that is the final answer

Now , Finding

1)

sin a = `1/(csc a )` = 1/2

2) cos a = `(+- sqrt(csc ^2 a -1)/ (csc a))`

= `(+- sqrt(2^2 -1)/ (2))`

= `(+- sqrt(4 -1)/ (2))`

= `(+- sqrt(3)/ (2))`

As the angle is in the Second quadrent **cos is negative** (see table II )

so **`cos a = (- sqrt(3)/ (2))` **

3)

`Tan a = (+- 1/(sqrt(csc^2 a -1)))`

= `(+- 1/(sqrt(2^2 -1)))`

= `(+- 1/(sqrt(4 -1)))`

=`(+- 1/(sqrt(3)))`

As the angle is in the Second quadrent **tan is negative** (see table II )

so ,

`tan a =(- 1/(sqrt(3)))`

simple :)

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