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

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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**

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)**

**Already Given that **

`tan(theta) = -15/8`

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

**Images:**