Express as a single sine or cosine function (note: this is using double angle formulas)
g) 8sin^2x-4
I just don't get this one. I know it's got something to do with the 1-2sin^2x double angle formula. It's the opposite though? :S
h) 1-2sin^2 (π/4-x/2)
= 1-sin^2(π/4-x/2)-sin^2(π/4-x/2)
= cos^2(π/4-x/2)-sin^2(π/4-x/2)
I got all the way up to cos (π/4 - x/2 + π/4 - x/2)
The answer is supposed to be sin x. I have no clue how they got that.
g) 8sin^2x-4
= 4(2sin^2 x - 1)
= 4(-cos 2x)
= -4cos 2x
h) 1-2sin^2 (π/4-x/2)
= cos 2(π/4-x/2)
= cos (π/2-x)
= cos(π/2)cosx + sin(π/2)sinx
= 0(sinx) + 1(sinx)
= sinx
Thanks a lot!
But can you please explain how you got 4(2sin^2 x - 1) for g and cos 2(π/4-x/2) for h?
I think I might understand h because 1-sin^2x = cosx but wouldn't it just be cos(π/4-x/2) cos(π/4-x/2)?
"But can you please explain how you got 4(2sin^2 x - 1)"
I took out a common factor of 4
"..and cos 2(π/4-x/2) for h"
Ok, let's work it in reverse.
You know that cos 2A = cos^2 A - sin^2 A, or cos 2A = 1 - 2sin^2 A ,right?
so I simply let A = (π/4-x/2)
then 2A = 2(π/4-x/2)
= π/2 - x
To express the given expression (8sin^2x - 4) as a single sine or cosine function, we can make use of the double angle formulas.
Here's how you can approach it:
g) 8sin^2x - 4
Step 1: Recall the double angle identity for sine: sin(2θ) = 2sinθcosθ.
Step 2: Rewrite sin^2x as (1 - cos^2x). This can be done using the Pythagorean identity: sin^2x + cos^2x = 1.
Step 3: Substitute the double angle identity and the expression from step 2 into the original equation.
8sin^2x - 4 = 8(1 - cos^2x) - 4 = 8 - 8cos^2x - 4 = -8cos^2x + 4
Step 4: Now, we can express -8cos^2x + 4 as a single cosine function using the identity: cos(2θ) = 2cos^2θ - 1.
Using this identity, we can rewrite the expression as:
-8cos^2x + 4 = -8cos(2x) + 4
So, the expression 8sin^2x - 4 can be expressed as -8cos(2x) + 4.
Moving on to part h:
h) 1 - 2sin^2(π/4 - x/2)
Step 1: Expand sin^2(π/4 - x/2) using the identity sin^2θ = 1 - cos^2θ.
1 - 2sin^2(π/4 - x/2) = 1 - 2(1 - cos^2(π/4 - x/2))
Step 2: Simplify the expression further.
1 - 2(1 - cos^2(π/4 - x/2)) = 1 - 2 + 2cos^2(π/4 - x/2) = -1 + 2cos^2(π/4 - x/2)
Step 3: Use the double angle identity for cosine: cos(2θ) = 2cos^2θ - 1.
-1 + 2cos^2(π/4 - x/2) = -1 + cos(2(π/4 - x/2))
Step 4: Simplify further.
-1 + cos(2(π/4 - x/2)) = cos(π/2 - x) = sin(x)
Thus, the expression 1 - 2sin^2(π/4 - x/2) can be simplified to sin(x).