Question

Find the other trigonometric functions if
if cos theta -3/2 and 180°< 0<270​

Answer 1

↗Let theta = A

Given

cos A = -3/2 where A lies in 3rd quadrant according to the equation 180° <A<270°

So we have to find all other trignometric angles that are :

sinA , tan A , cot A , sec A and cosec A

1 ) As we know that sec A is the reciprocal of cos A , so

cos A = -3/2

Then reciprocal of cos A will be the value of sec A

sec A = - 2/3

As sec A is also -ve in 3rd quadrant .

2) As we know that

$\sin(a) = \sqrt{1 - \cos( {a}^{2} ) }$

$\sin(a) = \sqrt{1 - {( \frac{ - 3}{2} )}^{2} }$

$\sin(a) = \sqrt{1 - \frac{9}{4} }$

$\sin(a) = - \frac{ \sqrt{5} }{2}$

As sin A also functions -ve in 3rd quadrant .

3) As we know that cosec is reciprocal of sin , so

sin A =- √5/2

Then Cosec will be reciprocal of it .

Cosec A = -2√5

As Cosec also functions-ve in 3rd quadrant.

4) As we know that

tan A = sin A / cos A so

We have already found the values of sin A and cos A ,so putting these values here

$\tan(a) = \frac{ - \frac{ \sqrt{5} }{2} }{ - \frac{3}{2} }$

tan A = √5/3

tan functions +ve in 3rd quadrant.

5) We know that cot is the reciprocal of tan , so

tan A = √5/3 then

cotA = 3/√5

As cot functions positive in 3rd quadrant.

hope it helps