If tan x=20/21, and π <x< 3π/2, what is cos⁡(x+π/2)?

Question

If tan x=20/21, and π < x < 3π/2, what is cos⁡(x+π/2)?

Answer 1

The domain restriction π < x < 3π/2 tells us that we are dealing with a point in the 3rd quadrant. Tangent is opposite/adjacent or y/x so the y = -20 and the x = -21 (negative because we are in Q3). Like this:

Using the Pythagorean Theorem we can solve for the hypotenuse "r": r = √(20² + 21²) = 29

Adding 90° (π/2 radians) essentially flips the triangle to be as follows (red triangle):

The cosine of the angle is adj/hyp = x/r = 20/29

Note that adding 90° to an angle in the 3rd Q will land you into the 4th Q where the cosine is positive.

Answer 2

Answer: -20/29 or approximately -0.69, the cosine is negative, positive 20/29

Here's the general formula: cos(A+B) = cosAcosB-sinAsinB

tanx=opposite side/adjacent side. cosx=adjacent side/hypotenuse. sinx=opposite side/hypotenuse

tanx=sinx/cosx

If tanx=20/21 then sinx/cosx=20/21

If adjacent side = 21 and opposite side=20, then the hypotenuse = sqr(21 squared + 20 squared)= sqr(441+400)=sqr(841)=29

sinx=-20/29 cosx=-21/29

Applying the general formula for cos(A+B) and substituting A=x, B=pi/2 gives:

cos(x+pi/2) = cosxcospi/2-sinxsinpi/2 sinpi/2=sin90=1 cospi/2=cos90=0

= cosx (0) - sinx (1) = -sinx

=sinx = 20/29 = approximately 0.69

x is in the 3rd quadrant pi<x<3pi/2 where tangent of x is always positive and cosine & sine always negative

when tanx is approximately 1, such as 20/21=0.95, it's about a 225 degree angle, or 5pi/4 radians if x is in the 3rd quadrant. tanx also = 1 in the 1st quadrant, but in this problem pi<x<3pi/2 or 180<x<270 degrees

the sin225=-0.707 which is close to -0.69

in the third quadrant, sinx is opposite side/hypotenuse which is a negative over positive. sinx<0

The actual angle x is about 5pi/4 randians or 180+45= 225 degrees

tan225=-1, and sin225=-0.707

sinx=-.0.69 which is close to -0.707 as x is close to 225 degrees