Use the trigonometric subtraction formula for sine to verify this identity:
sin((π / 2) – x) = cos x
First start by converting the angle in radians to degrees, hence will have
Sin(90-x)=cos x
=sin90cosx-cos90sinx, sin 90=1 and cos 90=0
=1(cos x)- 0(sin x)
=cos x (proved)
Newton's math is correct, but there is certainly no reason to convert to degrees.
To verify the given identity using the trigonometric subtraction formula for sine, we start with the formula:
sin(a - b) = sin(a) * cos(b) - cos(a) * sin(b).
In this case, let's substitute a with π/2 and b with x:
sin((π/2) - x) = sin(π/2) * cos(x) - cos(π/2) * sin(x).
Now, let's evaluate the individual trigonometric functions:
sin(π/2) = 1 (since sin(π/2) equals 1),
cos(π/2) = 0 (since cos(π/2) equals 0).
Substituting the values:
sin((π/2) - x) = 1 * cos(x) - 0 * sin(x).
sin((π/2) - x) = cos(x) - 0.
sin((π/2) - x) = cos(x).
Therefore, we have verified that sin((π/2) - x) = cos(x) using the trigonometric subtraction formula for sine.
To verify the identity sin((π / 2) – x) = cos x using the trigonometric subtraction formula for sine, we will need to express sin((π / 2) – x) in terms of cos x.
The trigonometric subtraction formula for sine states that sin(a - b) = sin a cos b - cos a sin b. In this case, a = π / 2 and b = x.
Let's substitute the values into the formula:
sin((π / 2) – x) = sin(π / 2) cos(x) - cos(π / 2) sin(x)
Working through each term:
sin(π / 2) = 1, as sin(π / 2) is the value of the sine function when the angle is a right angle.
cos(π / 2) = 0, as cos(π / 2) is the value of the cosine function when the angle is a right angle.
Substituting these values back into the equation:
sin((π / 2) – x) = 1 * cos(x) - 0 * sin(x)
sin((π / 2) – x) = cos(x) - 0
sin((π / 2) – x) = cos(x)
Therefore, sin((π / 2) – x) equals cos(x), and the identity is verified using the trigonometric subtraction formula for sine.