these must be written as a single trig expression, in the form sin ax or cos bx.
a)2 sin 4x cos4x
b)2 cos^2 3x-1
c)1-2 sin^2 4x
I need to learn this!! if you can show me the steps and solve it so I can learn I'd be grateful!!!
1) apply the formula for sin 2z
2)3) cos^2z + sin^2 z =1 in both of these. substitute this for the 1 in the equations.
Sin^2x-2sin2x-5cos^2alfa=0
a) To write it as a single trig expression, let's use the double angle identity for sine:
sin 2z = 2sin z cos z.
Therefore, let's express sin 4x cos 4x using the double angle identity:
sin 4x cos 4x = 2(sin 2(4x)/2) cos 2(4x)/2)
= 2sin 2(2x) cos 2(2x)
= 2(sin 2x cos 2x).
Now, let's replace the 2x with a variable, let's say a:
2(sin 2x cos 2x) = 2(sin a cos a).
So, the single trig expression for 2sin 4x cos 4x is 2sin a cos a, where a = 2x.
b) To write 2 cos^2 3x - 1 as a single trig expression, let's use the Pythagorean identity:
sin^2 z + cos^2 z = 1.
Since we have the expression cos^2 3x, we can express it as 1 - sin^2 3x:
2 cos^2 3x - 1 = 2(1 - sin^2 3x) - 1
= 2 - 2sin^2 3x - 1
= 1 - 2sin^2 3x.
So, the expression 2 cos^2 3x - 1 can be written as 1 - 2sin^2 3x.
c) Similarly, let's use the Pythagorean identity to rewrite this expression:
1 - 2sin^2 4x = 1 - 2(1 - cos^2 4x)
= 1 - 2 + 2cos^2 4x
= 2cos^2 4x - 1.
Therefore, the single trig expression for 1 - 2sin^2 4x is 2cos^2 4x - 1.
By following these steps and using the appropriate trigonometric identities, you can rewrite the given expressions as single trig expressions in the form sin ax or cos bx.