Write the expression as a single trigonometric functions:
cos 6x cosx - sin6x sin x
if you could do it step by step?
also... How do I prove that sin(90deg + theta) = cos(theta)??
also step by step so i may learn?
the important identity here is
cos(A+B) = cosAcosB - sinAsinB
So, it looks like you have
cos(6x+x) = cos 7x
You also know that
sin(A+B) = sinAcosB + cosAsinB
so,
sin(90+x) = sin90cosx + cos90sinx = 1*cosx = 0*sinx = cosx
Do lots of these, since they will come in handy later on.
Sure! Let's break down both problems step by step.
1. Writing the expression as a single trigonometric function:
To simplify the given expression, we can use the trigonometric identity called the product-to-sum formula, which states that cos A cos B - sin A sin B is equal to cos(A + B).
Let's apply this formula to the given expression:
cos 6x cos x - sin 6x sin x
Using the product-to-sum formula, we can rewrite it as:
= cos(6x + x)
= cos(7x)
So, the simplified expression is cos(7x).
2. Proving that sin(90° + θ) = cos θ:
To prove this trigonometric identity, we can use the angle addition formula for sine, which states that sin(A + B) = sin A cos B + cos A sin B.
Let's use this formula to prove the given identity step by step:
sin(90° + θ)
Using the angle addition formula, we can rewrite it as:
= sin 90° cos θ + cos 90° sin θ
Now, let's evaluate the sine and cosine of 90°:
sin 90° = 1
cos 90° = 0
Plugging in these values, we have:
= (1)(cos θ) + (0)(sin θ)
= cos θ + 0
= cos θ
Therefore, we have proved that sin(90° + θ) equals cos θ.
I hope that helps! Let me know if you have any further questions.