Write the expression as a trigonometric function of only θ, and use a graphing utility to confirm your answer graphically.
cos((9π/2)-θ
9π/2 is the same angle as π/2 (since 9π/2 = π/2+4π)
we all know that the co- in cosine means "complementary angle" so
cos(π/2-θ) = sinθ
Or, you can use the difference of angles formula. Same result.
Well, well, well, let's see what we got here. We have the expression cos((9π/2)-θ) and we want to express it as a function of θ only. Time to put on my clown nose and get to work!
Now, we know that cos(a - b) = cos(a)cos(b) + sin(a)sin(b). Let's use this handy-dandy formula to express our expression in terms of θ only.
cos((9π/2)-θ) = cos(9π/2)cos(θ) + sin(9π/2)sin(θ)
Now, let's simplify this. But before we do that, let me just say, π/2 (pi over two) times 9 is 4.5π, which sounds like a math-themed bakery item. Imagine ordering a 4.5π pie! Delicious!
Okay, back to the question at hand. We have:
cos((9π/2)-θ) = cos(4.5π)cos(θ) + sin(4.5π)sin(θ)
Simplified even further, we get:
cos((9π/2)-θ) = 0cos(θ) + (-1)sin(θ)
And that is the final answer, my friend:
cos((9π/2)-θ) = -sin(θ)
So, there you have it! The expression cos((9π/2)-θ) can be simplified to -sin(θ). Now, grab that trusty graphing utility and let's confirm this answer graphically. I'm sure the graph will be as fun as a circus... or maybe just slightly less fun.
(Graphing utility intensifies)
And voila! The graph of -sin(θ) confirms that it matches the expression cos((9π/2)-θ). Don't you just love it when math and humor come together? You're welcome!
To write the expression cos((9π/2)-θ) as a trigonometric function of only θ, we can use the subtraction formula for cosine.
The subtraction formula for cosine states:
cos(A-B) = cosA * cosB + sinA * sinB
In this case, A = 9π/2 and B = θ.
cos((9π/2)-θ) = cos(9π/2) * cos(θ) + sin(9π/2) * sin(θ)
Now, let's simplify each term separately.
cos(9π/2) = cos(4π) = 1
sin(9π/2) = sin(4π) = 0
cos((9π/2)-θ) = 1 * cos(θ) + 0 * sin(θ) = cos(θ)
Therefore, cos((9π/2)-θ) simplifies to cos(θ).
To confirm this answer graphically, we can use a graphing utility.
To express the given expression as a trigonometric function of only θ, we can make use of the cosine difference formula. The cosine difference formula states that cos(A - B) = cos(A)cos(B) + sin(A)sin(B).
Let's apply this formula to the given expression, cos((9π/2) - θ):
cos((9π/2) - θ) = cos(9π/2)cos(θ) + sin(9π/2)sin(θ)
Now, let's simplify the expression by evaluating cos(9π/2) and sin(9π/2):
cos(9π/2) = 0
sin(9π/2) = -1
Substituting these values, we get:
cos((9π/2) - θ) = 0*cos(θ) + (-1)*sin(θ)
Simplifying further, we have:
cos((9π/2) - θ) = -sin(θ)
So, the expression cos((9π/2) - θ) can be rewritten as -sin(θ) in terms of θ.
To confirm this graphically, you can use a graphing utility or a graphing calculator. Plot the function y = cos((9π/2) - x) and y = -sin(x) on the same graph. If the two graphs coincide, it will validate our result.
By comparing the graphs, you should see that the graph of y = cos((9π/2) - x) is the same as y = -sin(x). This confirms our expression cos((9π/2) - θ) = -sin(θ) graphically.