Sin(u)=12/13 and cos(u) is negative?

Question

sin(u+pi) =
cos(u+pi) =
tan(u+pi) =
sin(u+(pi/2)) =
cos(u+(pi/2)) =
tan(u+(pi/2)) =

 

I figured out the first few but this +pi stuff has got me confused

Answer 1

Hi Skylar,

Sin(u)=12/13 and cos(u) is negative

cos(u)< 0    in the range:  pi/2≤ u ≤3pi/2, 2^nd and 3^rd quadrants

sin(u) = opp/hyp = 12/13, cos(u) = adj/hyp, find adj

hyp² = opp² + adj²

13² = 12² + adj²

adj = ± 5

since cos()< 0, pick adj = - 5 ==>

cos(u) = - 5/13, tan(u) = opp/adj = -12/5

Also sin(pi) = 0, sin(pi/2)=1, cos(pi) = -1, cos(pi/2) =0.

tan(pi) = 0, tan(pi/2)=∞

 

sin(u+pi) = sin(u)cos(pi)+cos(u)sin(pi)

              = (12/13)(-1)+(-5/13)(0)

              = - 12/13

cos(u+pi) = cos(u)cos(pi)-sin(u)sin(pi)

               = (-5/13)(-1 )-(12/13)(0)

               = 5/13
tan(u+pi) = (tan(u)+tan(pi))/(1-tan(u)tan(pi)) 

               = (-12/5 + 0)/(1-(-12/5)(0))

               =-12/5
sin(u+(pi/2)) = sin(u)cos(pi/2)+cos(u)sin(pi/2)

                    =(12/13)(0)+(-5/13)(1)

                    = -5/13
cos(u+(pi/2)) = cos(u)cos(pi/2)-sin(u)sin(pi/2)

                    = (-5/13)(0)-(12/13)(1)

                    =-12/13
tan(u+(pi/2)) = sin(u+(pi/2))/cos(u+(pi/2)) 

                    =(-5/13)/(-12/13)

                    =5/12

Answer 2

Hey Skylar,

 

Write the formulas first:

 

sin (u + v) = sin u cos v + sin v cos u

 

cos (u + v) = cos u cos v - sin u sin v

 

tan (u + v) = (tan u + tan v) / [1 - (tan u) (tan v)]

 

 

 

sin u = 12/13.  Then, cos u can be either 5/13 or -5/13.

 

sin v = 5/13  and  cos v = -12/13.

 

 

sin (u + pi) = sin u cos pi + sin pi cos u

 

(12/13) (-1) + (0) (-5/13) = -12/13 = sin (u + pi)

cos (u + pi) = cos u cos pi - sin u sin pi

(-5/13) (-1) - (12/13) (0) = 5/13 = cos (u + pi)

tan (u + pi) = (tan u + tan pi) / [1 - (tan u) (tan pi)]   Note: tan = sin / cos

 

(-2.4 + 0) / [ 1 - (-2.4)(0)] = -2.4 / 1 = -2.4 = tan (u + pi).

 

 

sin u = 12/13. Then, cos u can be either 5/13 or -5/13.

sin v = 5/13 and cos v = -12/13.

tan u = sin u / cos u = (12/13) / (-5/13) = -2.4

 

sin (u + pi/2) = sin u cos pi/2 + sin pi/2 cos u

 

(12/13) (0) + (1) (-5/13) = 0 - 5/13 = -5/13 = sin (u + pi/2).

 

 

cos (u + pi/2) = cos u cos pi/2 - sin u sin pi/2

 

(-5/13) (0) - (12/13) (1) = 0 - 12/13 = -12/13 = cos (u + pi/2).

 

 

tan (u + pi/2) = (tan u + tan pi/2) / [1 - (tan u) (tan pi/2)]     Note: tan = sin / cos

 

tan u = sin u / cos u = (12/13) / (-5/13) = -2.4 = tan (u + pi/2)  

 

note: tan pi/2 = und or ∞

 

 

(-2.4) + (∞) / [ 1 - (-2.4) (∞) ] = (-2.4) + (∞) / [ 1 + (2.4) (∞) ] = ∞ or undefined.

 

I hope you understood.

 

Good luck to you.