What will be the maximum value of 2sin theta + 4cos theta ?

Question

Answer

Maximum Value of 2sinθ 4cosθ

When we are asked to find the maximum value of an expression, it means we need to find the highest possible output of that expression. In this case, we need to find the maximum value of the expression 2sinθ 4cosθ.

Using Trigonometric Identities

One way to find the maximum value of the expression is to use trigonometric identities. We can rewrite the expression as:

2sinθ 4cosθ = 2(sinθ 2cosθ)

We can then use the following identity:

sin2θ + cos2θ = 1

Rearranging this identity, we get:

sin2θ = 1 - cos2θ

Substituting this into our expression, we get:

2(sinθ 2cosθ) = 2(sinθ 2√(1 - sin2θ))

Now, we can treat this expression as a function of sinθ. Let f(sinθ) = 2(sinθ 2√(1 - sin2θ)). We want to find the maximum value of f(sinθ) for 0 ≤ θ ≤ 2π.

To find the maximum value of f(sinθ), we need to find its critical points. We can do this by taking the derivative of f(sinθ) with respect to sinθ:

f'(sinθ) = 2cosθ - 4sinθ√(1 - sin2θ)

Setting f'(sinθ) = 0, we get:

cosθ = 2sinθ√(1 - sin2θ)

Squaring both sides, we get:

cos2θ = 4sin2θ(1 - sin2θ)

Expanding both sides, we get:

cos2θ = 4sin2θ - 4sin4θ

Using the identity sin2θ + cos2θ = 1, we can rewrite this as:

1 - sin2θ = 4sin2θ - 4sin4θ

Bringing all terms to one side, we get:

4sin4θ - 5sin2θ + 1 = 0

This is a quadratic equation in sin2θ. Solving for sin2θ, we get:

sin2θ = 1/4 or sin2θ = 1

Since 0 ≤ θ ≤ 2π, the first solution gives us sinθ = ±1/2 and the second solution gives us sinθ = ±1.

Finding the Maximum Value

Now that we have found the critical points of f(sinθ), we need to determine which one gives us the maximum value. We can do this by evaluating f(sinθ) at the critical points and at the endpoints of the interval 0

Answer

4

Answer

-6

Answer

I suppose 4 will be answer because maximum value of cos is 1which will arrive when theta =0

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