1. For what values is the following equation true? sin(x+20)=cos(y−10)
A. X=30 and y=60
B. X=30 and y=40
C. X=20 and y=50
D. X=20 and y=60
To solve this equation, we can use the trigonometric identities:
sin(x+20) = cos(y-10)
sin(x+20) = sin(90 - (y-10))
Since sin(a) = sin(b) implies either a = b or a + b = 180 degrees, we have:
x + 20 = 90 - (y-10)
x + 20 = 90 - y + 10
x + 20 = 100 - y
x = 100 - y - 20
x = 80 - y
Now, substituting the values given in the answer choices we can find the right answer:
A. x=30 and y=60
x = 80 - 60
x = 20
This does not work because sin(20+20) does not equal cos(60-10).
B. x=30 and y=40
x = 80 - 40
x = 40
This does not work because sin(40+20) does not equal cos(40-10).
C. x=20 and y=50
x = 80 - 50
x = 30
This does not work because sin(30+20) does not equal cos(50-10).
D. x=20 and y=60
x = 80 - 60
x = 20
This works because sin(20+20) = sin(40) and cos(60-10) = sin(50).
Therefore, the correct answer is D. X=20 and y=60.