# function value: `tan(theta) = -15/8` constraint: `sin(theta) > 0`   Find the values of the six trigonometric functions of theta with the given constraint.

balajia | Student

tan(theta) = -15/8

so,

1) Cot (theta) = -8/15

tan(theta) = -15/8

=> tan(pi - theta) = 15/8   [see the attachment ]

tan(pi - theta) = (AB)/(BO)

so AO = sqrt(AB^2 + BO^2)

= sqrt(15^2 + 8^2)

= sqrt(289)

= 17

2) sin(pi - theta) = (AB)/(AO) = 15 /17

=> sin(theta) = 15 /17

cosec(theta) = 17/15

3) cos(pi - theta) = (BO)/(AO) = 8 /17

=>cos( theta) = -8 /17

sec( theta) = -17/8

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kspcr111 | Student

Given

`tan(theta) = -15/8`

and given

`sin(theta) > 0`

=> the interval of sine is [0,1]  and the value of `tan (theta)` is negative, so the theta is in the second quadrant .

so now finding all trigonometric functions we get as follows:

By the attachments given below, we can easily find the values, please use them for reference.

1) `sin (theta ) = (+- tan(theta))/(sqrt(1+ tan^2(theta)))`

= `(+- (15/8))/(sqrt(1+ (15/8)^2))`

= `(+- 15/17)`

as Theta is in the second quadrant so `sin (theta)` is positive

so   `sin (theta) =( 15/17)`

=> `cosec (theta) = (17/15)`

2)

`cos (theta) = (+- 1)/(sqrt(1+ tan^2(theta)))`

` =(+- 1)/(sqrt(1+ (15/8)^2))`

`= (+- 8/17)`

as Theta is in the second quadrant so `cos (theta)` is negative

so  ` cos(theta) =( -8/17)`

=> `sec (theta) = (-17/8)`

3)

`tan(theta) = -15/8`

so `cot (theta) = -8/15`

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