Maximum and minimum value of Sin^2(60-x)+sin^2(60+x)

To find the maximum and minimum values of the expression Sin²(60° - x) + Sin²(60° + x), we can start by simplifying the trigonometric terms involved. This expression involves sine functions, which have certain properties that can help us analyze the problem more efficiently.

Breaking Down the Expression

First, let's utilize the angle addition and subtraction identities for sine. According to these identities:

• Sin(α ± β) = Sin(α)Cos(β) ± Cos(α)Sin(β)

Applying this to our case:

• Sin(60° - x) = Sin(60°)Cos(x) - Cos(60°)Sin(x)
• Sin(60° + x) = Sin(60°)Cos(x) + Cos(60°)Sin(x)

Calculating Sin² Terms

Now we know that Sin(60°) = √3/2 and Cos(60°) = 1/2. Therefore, we can rewrite the sine terms:

• Sin(60° - x) = (√3/2)Cos(x) - (1/2)Sin(x)
• Sin(60° + x) = (√3/2)Cos(x) + (1/2)Sin(x)

Next, we square these terms:

• Sin²(60° - x) = [(√3/2)Cos(x) - (1/2)Sin(x)]²
• Sin²(60° + x) = [(√3/2)Cos(x) + (1/2)Sin(x)]²

Expanding the Squares

By expanding these two squares, we can combine them into a single expression:

• Sin²(60° - x) = (3/4)Cos²(x) - √3/2 Cos(x)Sin(x) + (1/4)Sin²(x)
• Sin²(60° + x) = (3/4)Cos²(x) + √3/2 Cos(x)Sin(x) + (1/4)Sin²(x)

When we add these two results together, the cross terms will cancel out:

• Sin²(60° - x) + Sin²(60° + x) = (3/4)Cos²(x) + (3/4)Cos²(x) + (1/4)Sin²(x) + (1/4)Sin²(x)
• = (3/2)Cos²(x) + (1/2)Sin²(x)

Using Trigonometric Identities

To further simplify this expression, we can use the identity Sin²(x) + Cos²(x) = 1:

• Sin²(x) = 1 - Cos²(x)

Substituting this into our expression gives:

• (3/2)Cos²(x) + (1/2)(1 - Cos²(x))

Which simplifies to:

• =(3/2)Cos²(x) + (1/2) - (1/2)Cos²(x) = Cos²(x) + (1/2)

Finding Maximum and Minimum Values

Now we have a simpler expression: Cos²(x) + 1/2. The value of Cos²(x) varies between 0 and 1, as Cos(x) oscillates between -1 and 1.

• Minimum value of Cos²(x) = 0, giving us 0 + 1/2 = 1/2.
• Maximum value of Cos²(x) = 1, resulting in 1 + 1/2 = 3/2.

Conclusion: Summary of Results

Thus, the maximum value of Sin²(60° - x) + Sin²(60° + x) is 3/2, and the minimum value is 1/2. Understanding these properties of sine and cosine functions allows us to manipulate and analyze trigonometric expressions effectively!