4. 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).
Well, well, well, look who's in the mood for some trigonometry jokes! I must say, Rory's attempt to simplify sin(180-è) is a little off the mark.
You see, Rory thought that sin(180-è) could be simplified to sin 180 - sin è. But unfortunately, that's not the case. Sin 180 is indeed zero, but subtracting sin è from zero won't give us −sin è.
So, how can we correctly express sin(180-è) in a simpler form? Well, let me enlighten you with a little trigonometric magic. It turns out that sin(180-è) is the same as sin(è). It's like a sneaky switcheroo!
As for cos(180-è), we can use similar trickery. Cos(180-è) is equal to -cos(è). The cos value goes all negative on us, but that's just how life is sometimes.
Remember, math is all about finding the right angle and having fun along the way! Keep those mathematical quips coming, my friend!
Rory's solution for simplifying sin(180−è) is not correct. The correct way to express sin(180−è) in simpler form is using the angle subtraction formula for sine:
sin(180−è) = sin(180)cos(è) − cos(180)sin(è)
Since sin(180) = 0 and cos(180) = -1, the expression simplifies to:
sin(180−è) = 0(cos(è)) − (-1)(sin(è)) = sin(è)
So, the correct answer is sin(180−è) simplifies to sin(è).
Similarly, for cos(180−è), we can use the angle subtraction formula for cosine:
cos(180−è) = cos(180)cos(è) + sin(180)sin(è)
Since cos(180) = -1 and sin(180) = 0, the expression simplifies to:
cos(180−è) = -1(cos(è)) + 0(sin(è)) = -cos(è)
Therefore, the correct way to express cos(180−è) in simpler form is -cos(è).
To simplify the expression sin(180−è) correctly, we need to review the trigonometric identities.
First, let's consider the formula for the sine of a difference of two angles:
sin(A − B) = sin A cos B − cos A sin B
Now let's apply this formula to the expression sin(180 − è):
sin(180 − è) = sin 180 cos è − cos 180 sin è
However, we know that sin 180 is equal to 0 and cos 180 is equal to -1. Therefore, we can substitute these values:
sin(180 − è) = 0 * cos è − (-1) * sin è
Simplifying further:
sin(180 − è) = -sin è
So the expression sin(180 − è) simplifies to -sin è, which verifies Rory's solution and confirms that it is correct.
Now let's move on to the second part of the question, simplifying the expression cos(180 − è). We'll use the formula for the cosine of a difference of two angles:
cos(A − B) = cos A cos B + sin A sin B
Applying this formula to cos(180 − è):
cos(180 − è) = cos 180 cos è + sin 180 sin è
Again, we can substitute the values cos 180 = -1 and sin 180 = 0:
cos(180 − è) = -1 * cos è + 0 * sin è
Simplifying further:
cos(180 − è) = -cos è
So the expression cos(180 − è) simplifies to -cos è. This means Rory's solution is also correct for cos(180 − è).
Isn't
Sin(A+B)=SinA Cos B + CosA sinB
sin(180-e)= Sin180 Cos -e + Cos180 sin -e
= 0 -sin -e=sine