if sin6A=cos9A, then angle A is equal to?
1 6
2 36
3 54
4 1 half
if sin x = cos y, then x and y must be complementary angles, that is, they must add up to 90º
so 6A + 9A = 90
A = 6
as Emileee stated, just thought I would explain why
angle A = 6
Angle A = 6
To find the value of angle A, we can use the trigonometric identity: sin^2(A) + cos^2(A) = 1.
Given that sin(6A) = cos(9A), we can rewrite this equation as sin^2(6A) + sin^2(9A) = 1.
Since sin^2(A) = 1 - cos^2(A), we can substitute and rewrite the equation as (1 - cos^2(6A)) + (1 - cos^2(9A)) = 1.
Simplifying further, we have 2 - cos^2(6A) - cos^2(9A) = 1.
Rearranging the equation, we get cos^2(6A) + cos^2(9A) = 1.
Since cos^2(A) = 1 - sin^2(A), we can substitute and rewrite the equation as cos^2(6A) + (1 - sin^2(9A)) = 1.
Simplifying further, we have cos^2(6A) + 1 - sin^2(9A) = 1.
Rearranging the equation, we get sin^2(9A) = cos^2(6A).
Since sin^2(A) = 1 - cos^2(A), we can substitute and rewrite the equation as 1 - cos^2(9A) = 1 - cos^2(6A).
After canceling out the common terms, we have cos^2(9A) = cos^2(6A).
To solve this equation, we have two possibilities:
1) cos(9A) = cos(6A)
In this case, the angles 9A and 6A have the same cosine value. Two angles have the same cosine value if they differ by a multiple of 360 degrees (or 2π radians).
Therefore, we can write 9A - 6A = 360n, where n is an integer.
Simplifying, we have 3A = 360n, which leads to A = 120n, where n is an integer.
2) cos(9A) = -cos(6A)
In this case, the angles 9A and 6A have opposite cosine values. Two angles have opposite cosine values if they differ by a multiple of 180 degrees (or π radians).
Therefore, we can write 9A - 6A = 180n, where n is an integer.
Simplifying, we have 3A = 180n, which leads to A = 60n, where n is an integer.
Based on the given options, the possible values for angle A are 60 degrees (option 2) and 120 degrees (option 3).