1.Using exact values, verify that
cos 60°=1-2 sin² 30°
I am not certain what I am seeing.
Is it
cos^2 60=1-2sin^2 30 ?
3/4=?=1-2*1/4
so it doesn't verify...so I must not be seeing the type clearly
This is exacly how the question goes, This is a test question.
1.Using exact values, verify that
cos 60°=1-2 sin² 30°
To verify the given equation, we need to express both sides in terms of exact values.
Let's start by evaluating the right-hand side (RHS) of the equation. We'll use the identities:
sin²θ = (1/2)(1 - cos 2θ) [double-angle identity for cosine]
cos 2θ = 1 - 2sin²θ [double-angle identity for sine]
First, substitute θ = 30° into the second identity:
cos 60° = 1 - 2sin² 30°
Next, substitute θ = 30° into the first identity:
sin² 30° = (1/2)(1 - cos 60°)
Now, let's find the values of sin 30° and cos 60° to verify the equation.
sin 30° = 1/2 [exact value]
cos 60° = 1/2 [exact value]
Substitute these values into the equation:
1/2 = 1 - 2(1/2)(1 - cos 60°)
Simplifying further:
1/2 = 1 - (1 - cos 60°)
1/2 = 1 - 1 + cos 60°
1/2 = cos 60°
The left-hand side (LHS) is equal to the right-hand side (RHS). Therefore, cos 60° = 1 - 2sin² 30° is verified using exact values.