Final answer:
The exact value of sin 1400 degrees cos 20 degrees - cos 140 degrees sin 20 degrees is approximately 0.342.
Step-by-step explanation:
The given expression is sin 1400 degrees cos 20 degrees - cos 140 degrees sin 20 degrees.
Using the trigonometric identities, we can rewrite this expression:
sin 1400 cos 20 - cos 140 sin 20
sin (a + b) = sin a cos b + cos a sin b, so:
sin 1400 cos 20 - cos 140 sin 20 = sin (1400 + 20)
The sine function has a period of 360 degrees, so we can subtract 7 periods (7 * 360 degrees) to get an angle within one period:
1400 + 20 - 7 * 360 = 20 degrees
Therefore, the exact value of the given expression is sin 20 degrees, which is approximately 0.342.