Suppose
cos(u)=3/5
and sin(u) is positive.
sin(u)=
sin(u−π)=
cos(u−π)=
sin(u−π/2)=
cos(u−π/2)=
The co- in cosine means "of the complementary angle." So, cos(x) = sin(π/2-x)
Here,
sin(u) = 4/5
Now just recall the formulas for sum of angles, and you get
sin(u-π) = -sin(u)
cos(u-π) = -cos(u)
sin(u-π/2) = -cos(u)
cos(u-π/2) = sin(u)
ty!
To find the values of sin(u), sin(u-π), cos(u-π), sin(u-π/2), and cos(u-π/2), we will use trigonometric identities and the given information that cos(u) = 3/5 and sin(u) is positive.
First, let's start with sin(u). We can use the Pythagorean identity to find sin(u) given the value of cos(u):
sin^2(u) + cos^2(u) = 1
Given cos(u) = 3/5, we can solve for sin(u):
sin^2(u) + (3/5)^2 = 1
sin^2(u) + 9/25 = 1
sin^2(u) = 1 - 9/25
sin^2(u) = 16/25
sin(u) = ± √(16/25)
sin(u) = ± 4/5
Since sin(u) is given to be positive, we can conclude that sin(u) = 4/5.
Moving on to sin(u-π), we can use the angle subtraction formula for sine:
sin(u-π) = sin(u)cos(π) - cos(u)sin(π)
Since cos(π) = -1 and sin(π) = 0, the formula simplifies to:
sin(u-π) = sin(u) * (-1) - cos(u) * 0
sin(u-π) = -sin(u)
sin(u-π) = -4/5
Next, let's find cos(u-π). Again, using the angle subtraction formula, this time for cosine:
cos(u-π) = cos(u)cos(π) + sin(u)sin(π)
Since cos(π) = -1 and sin(π) = 0, the formula simplifies to:
cos(u-π) = cos(u) * (-1) + sin(u) * 0
cos(u-π) = - cos(u)
cos(u-π) = -3/5
Moving on to sin(u-π/2), using the angle subtraction formula for sine:
sin(u-π/2) = sin(u)cos(π/2) - cos(u)sin(π/2)
Since cos(π/2) = 0 and sin(π/2) = 1, the formula simplifies to:
sin(u-π/2) = sin(u) * 0 - cos(u) * 1
sin(u-π/2) = - cos(u)
sin(u-π/2) = -3/5
Finally, let's find cos(u-π/2). Using the angle subtraction formula for cosine:
cos(u-π/2) = cos(u)cos(π/2) + sin(u)sin(π/2)
Since cos(π/2) = 0 and sin(π/2) = 1, the formula simplifies to:
cos(u-π/2) = cos(u) * 0 + sin(u) * 1
cos(u-π/2) = sin(u)
cos(u-π/2) = 4/5
To summarize:
sin(u) = 4/5
sin(u-π) = -4/5
cos(u-π) = -3/5
sin(u-π/2) = -3/5
cos(u-π/2) = 4/5