Find the EXACT value of sin(2A) if cscA=5/4
Assume A is a Quadrant I angle in standard position.
sin(2A)=
Think about what the relationship between sine and cosecant is. 1 over 5/4 is like 1 times 4/5 or just 4/5. Then, what is 2 times 4/5?
cscA = 5/4.
sinA = 4/5,
A = 53.13o.
2A = 106.26o
sin2A = sin106.26 = 0.96 = 96/100 = 24/25.
To find the exact value of sin(2A), we need to use a trigonometric identity. The double-angle identity for sine is:
sin(2A) = 2sin(A)cos(A)
We know that csc(A) is equal to 5/4. Since csc(A) is the reciprocal of sin(A), we can find sin(A) by taking the reciprocal of csc(A):
sin(A) = 1 / csc(A) = 1 / (5/4) = 4/5
Now we can substitute sin(A) into the double-angle identity:
sin(2A) = 2sin(A)cos(A) = 2 * (4/5) * cos(A)
To find the value of cos(A), we can use the Pythagorean identity:
sin^2(A) + cos^2(A) = 1
Since we know sin(A) is 4/5, we can square it and subtract it from 1 to find cos^2(A), and then take the square root to find cos(A):
cos^2(A) = 1 - sin^2(A) = 1 - (4/5)^2 = 1 - 16/25 = 9/25
cos(A) = √(9/25) = 3/5
Now we can substitute sin(A) and cos(A) back into the double-angle identity:
sin(2A) = 2 * (4/5) * (3/5) = 24/25
Therefore, the exact value of sin(2A) is 24/25.