Explain why the value of [cos θ,sin θ]• [cos(90 + θ),sin(90 + θ)] is independent of θ.
because cos(90+θ) = -sinθ and sin(90+θ) = cosθ
Ah, the mystical world of mathematics! Let's unravel this cosmic question with a dose of humor, shall we?
Imagine you have a pair of cosmic penguins, Mr. Cosine (cos θ) and Mrs. Sine (sin θ), frolicking together. They love to explore strange new angles.
Now, one day they stumble upon another adventurous pair called Mr. Hipster (cos(90 + θ)) and Mrs. Rad (sin(90 + θ)). These two fancy ones like to marathon-run at right angles.
When Mr. Cosine and Mrs. Sine meet Mr. Hipster and Mrs. Rad, they decide to have a little "dot product" party.
Now, the dot product of two vectors is kind of like penguins hugging. It's all about finding the love between them. The more love (or dot product), the closer they are.
But here's the magic: no matter what angle θ they start at, the love between the two pairs of penguins never changes! It remains constant. They're like a mathematically immortal love story!
So, in simpler terms, the value of [cos θ,sin θ] • [cos(90 + θ),sin(90 + θ)] is independent of θ because the love between the vectors doesn't care about the angle, it's just constant. Love knows no bounds, even in mathematics!
Hope that brings a smile to your face, or at least tickles your mathematical sense of humor!
To prove that the value of the expression [cos θ, sin θ] • [cos(90 + θ), sin(90 + θ)] is independent of θ, we need to evaluate it and show that the result remains constant regardless of the value of θ.
First, let's compute the dot product of the two vectors:
[cos θ, sin θ] • [cos(90 + θ), sin(90 + θ)]
The dot product is calculated by multiplying corresponding components of the vectors and summing them up:
[cos θ * cos(90 + θ), sin θ * sin(90 + θ)] + [sin θ * cos(90 + θ), sin θ * sin(90 + θ)]
Using trigonometric identities, we can simplify this expression:
[cos θ * cos 90° - sin θ * sin 90°, sin θ * cos 90° + sin θ * sin 90°]
Applying the values of the trigonometric functions at 90°:
[cos θ * 0 - sin θ * 1, sin θ * 0 + sin θ * 1]
[-sin θ, sin θ]
As we can see, the result of the dot product is [-sin θ, sin θ] which is only dependent on θ through the sign of sin θ. Therefore, the value remains the same regardless of the value of θ.
To understand why the value of the expression [cos θ, sin θ] • [cos(90 + θ), sin(90 + θ)] is independent of θ, let's break it down step by step.
1. First, let's expand the expression [cos θ, sin θ] • [cos(90 + θ), sin(90 + θ)] using the dot product formula:
[cos θ, sin θ] • [cos(90 + θ), sin(90 + θ)] = (cos θ * cos(90 + θ)) + (sin θ * sin(90 + θ))
2. Next, let's simplify the expression by using trigonometric identities. Recall the identities:
a) cos(90 + θ) = -sin θ
b) sin(90 + θ) = cos θ
Substituting these values in the expression, we have:
[cos θ, sin θ] • [cos(90 + θ), sin(90 + θ)] = (cos θ * (-sin θ)) + (sin θ * cos θ)
3. Now, let's simplify further using the commutative property of multiplication:
(cos θ * (-sin θ)) + (sin θ * cos θ) = -sin θ * cos θ + sin θ * cos θ
4. Notice that the two terms -sin θ * cos θ and sin θ * cos θ are equal but opposite in sign, so they cancel each other out:
-sin θ * cos θ + sin θ * cos θ = 0
5. Thus, we're left with the value of 0. Regardless of the value of θ, the result of the expression [cos θ, sin θ] • [cos(90 + θ), sin(90 + θ)] is always 0.
Therefore, we can conclude that the value of [cos θ, sin θ] • [cos(90 + θ), sin(90 + θ)] is independent of θ.