6. The equation

sin(θ/2) [sin(θ/2) + cos(θ/2)] = β

has a solution, where β is a natural number. Then β is ______.

Find β in the Equation sin(θ/2)(sin(θ/2) + cos(θ/2)) = β

Understanding the Given Trigonometric Equation

The given equation contains a product involving sine and cosine functions. We are not directly asked to find the value of θ. Instead, the question asks us to determine the natural number β for which the equation has at least one real solution.

This means that we need to determine the range of the expression on the left-hand side. Once we know all possible values that the trigonometric expression can take, we can identify which natural number lies within that range.

The given equation is:

sin(θ/2) [sin(θ/2) + cos(θ/2)] = β

To simplify the calculation, let:

x = θ/2

The equation then becomes:

sin x (sin x + cos x) = β

Step-by-Step Solution

Step 1: Expand the Trigonometric Expression

Expanding the product on the left-hand side gives:

β = sin²x + sin x cos x

Now we need to determine the range of the expression sin²x + sin x cos x. For this purpose, we use standard trigonometric identities that convert squared and product terms into expressions involving a single angle.

Step 2: Apply Standard Trigonometric Identities

We use the following identities:

sin²x = (1 − cos 2x)/2

and

sin x cos x = (sin 2x)/2

Substituting these identities into the expression for β:

β = (1 − cos 2x)/2 + (sin 2x)/2

Combining the two fractions:

β = [1 + sin 2x − cos 2x]/2

Therefore:

β = 1/2 + (sin 2x − cos 2x)/2

Step 3: Find the Range of sin 2x − cos 2x

A trigonometric expression of the form a sin t + b cos t has its values in the interval:

−√(a² + b²) ≤ a sin t + b cos t ≤ √(a² + b²)

For the expression:

sin 2x − cos 2x

the coefficient of sin 2x is 1, while the coefficient of cos 2x is −1. Therefore, its maximum possible magnitude is:

√[1² + (−1)²] = √2

Hence:

−√2 ≤ sin 2x − cos 2x ≤ √2

Step 4: Determine the Range of β

We obtained:

β = [1 + sin 2x − cos 2x]/2

Since the minimum value of sin 2x − cos 2x is −√2 and its maximum value is √2, the range of β is:

(1 − √2)/2 ≤ β ≤ (1 + √2)/2

Using the approximate value:

√2 ≈ 1.414

the lower limit becomes:

(1 − 1.414)/2 ≈ −0.207

and the upper limit becomes:

(1 + 1.414)/2 ≈ 1.207

Therefore, the complete range of β is:

−0.207 ≤ β ≤ 1.207

Identifying the Natural Number β

The question clearly states that β is a natural number. The positive natural numbers are:

1, 2, 3, 4, …

However, the trigonometric expression can take values only between approximately −0.207 and 1.207. Among all positive natural numbers, only the number 1 lies within this interval.

Therefore:

β = 1

Verification That β = 1 Gives a Solution

It is useful to verify that the equation actually has a real solution when β = 1. Substituting β = 1 into the simplified expression gives:

[1 + sin 2x − cos 2x]/2 = 1

Multiplying both sides by 2:

1 + sin 2x − cos 2x = 2

Therefore:

sin 2x − cos 2x = 1

This equation has real solutions. For example, if 2x = π/2, then:

sin(π/2) − cos(π/2) = 1 − 0 = 1

Thus, β = 1 is not only within the possible range of the expression but also produces an actual real solution of the given equation.

Why the Range Method Is the Best Approach

The central idea in this problem is that β must be a value that the left-hand side can actually attain. Instead of attempting to solve the equation separately for every possible natural number, determining the range immediately restricts all possible values of β.

The expression is transformed into a standard sine-cosine form, whose maximum and minimum values can be calculated directly. This makes the solution systematic and mathematically rigorous.

Final Answer

The only natural number in the range of the given trigonometric expression is 1. Therefore, β = 1.