How do you find the exact value of #cos[ arctan (-4/3) - arcsin (3/5) ]#?

Question

Asher Friedman · Accepted Answer

Zero.

You may use calculator #tan x = -4/3# --> #x = 126.86# (or #-53.14#) #sin y = 3/5# --># y = 36^@86# #x - y = 90^@# #cos 90^@ = 0#

Caroline Adams · Answer

undefined To find the exact value of cos[arctan(-4/3) - arcsin(3/5)], you would use trigonometric identities and properties. First, you would find the values of arctan(-4/3) and arcsin(3/5) using the given information. Then, you would use the properties of trigonometric functions to simplify the expression and find the cosine of the resulting angle. Here are the steps:

1. Find the values of arctan(-4/3) and arcsin(3/5):
   arctan(-4/3) = -53.13° (approximately)
   arcsin(3/5) = 36.87° (approximately)

2. Substitute these values into the expression:
   cos[-53.13° - 36.87°]

3. Use the cosine difference identity: cos(a - b) = cos(a)cos(b) + sin(a)sin(b):
   cos[-53.13°]cos[36.87°] + sin[-53.13°]sin[36.87°]

4. Substitute the cosine and sine values:
   cos[-53.13°] ≈ 0.6, sin[36.87°] ≈ 0.6
   cos[36.87°] ≈ 0.8, sin[-53.13°] ≈ -0.8

5. Calculate:
   ≈ (0.6)(0.8) + (-0.8)(0.6)
   ≈ 0.48 - 0.48
   ≈ 0

Therefore, the exact value of cos[arctan(-4/3) - arcsin(3/5)] is 0.