Asked to simplify the expression sin(180−è), Rory volunteered the following solution:
sin(180−è) = sin 180−sin è, and, because sin 180 is zero, it follows that sin(180−è) is the same as −sin è. Is this answer correct? If not, what is a correct way to express sin(180−è)
in simpler form? Answer the same question for cos(180 − è).
The solution provided by Rory is not correct. When simplifying trigonometric expressions, it is important to use the appropriate trigonometric identities and rules.
To correctly simplify the expression sin(180° - θ), we can use the angle subtraction identity for sine:
sin(A - B) = sin A cos B - cos A sin B
Using this identity, we have:
sin(180° - θ) = sin 180° cos θ - cos 180° sin θ
sin 180° is 0 and cos 180° is -1. Therefore, the expression simplifies to:
sin(180° - θ) = 0 * cos θ - (-1) * sin θ
= -sin θ
So, the correct way to express sin(180° - θ) in simpler form is simply -sin θ.
Now let's consider cos(180° - θ). We can use the angle subtraction identity for cosine:
cos(A - B) = cos A cos B + sin A sin B
Using this identity, we have:
cos(180° - θ) = cos 180° cos θ + sin 180° sin θ
cos 180° is -1 and sin 180° is 0. Therefore, the expression simplifies to:
cos(180° - θ) = -1 * cos θ + 0 * sin θ
= -cos θ
So, the correct way to express cos(180° - θ) in simpler form is simply -cos θ.