Sin^3x+cos^3x+3sinxcosx-1=0.find x

Please solve this question

To solve the equation \( \sin^3 x + \cos^3 x + 3 \sin x \cos x - 1 = 0 \), we can use a combination of algebraic identities and trigonometric properties. This type of equation can often be approached by recognizing patterns or using substitutions that simplify the expression.

Step 1: Recognizing the Form

The left-hand side of the equation contains the terms \( \sin^3 x \) and \( \cos^3 x \). We can apply the identity for the sum of cubes, which states that \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). Here, let \( a = \sin x \) and \( b = \cos x \). This gives us:

• Sum of cubes: \( \sin^3 x + \cos^3 x = (\sin x + \cos x)(\sin^2 x - \sin x \cos x + \cos^2 x) \)

Since \( \sin^2 x + \cos^2 x = 1 \), we can rewrite this as:

\( \sin^3 x + \cos^3 x = (\sin x + \cos x)(1 - \sin x \cos x) \)

Step 2: Rewriting the Equation

Now substituting this back into the original equation, we have:

\( (\sin x + \cos x)(1 - \sin x \cos x) + 3 \sin x \cos x - 1 = 0 \)

Expanding this gives:

\( (\sin x + \cos x)(1 - \sin x \cos x) + 3 \sin x \cos x - 1 = 0 \)

Step 3: Simplifying Further

Let's distribute \( (\sin x + \cos x)(1 - \sin x \cos x) \):

• \( \sin x + \cos x - \sin x \cos x (\sin x + \cos x) + 3 \sin x \cos x - 1 = 0 \)

This simplifies to:

\( \sin x + \cos x + (3 - \sin x \cos x)(\sin x + \cos x) - 1 = 0 \)

Step 4: Setting Up the Equation

Let’s denote \( y = \sin x + \cos x \). Then, \( \sin x \cos x = \frac{1}{2} \sin 2x \), and we can express the equation in terms of \( y \):

\( y + (3 - \frac{1}{2} \sin 2x)y - 1 = 0 \)

Step 5: Finding Solutions

From here, we can substitute \( \sin 2x \) in terms of \( y \) using the identities:

\( y^2 = \sin^2 x + \cos^2 x + 2 \sin x \cos x = 1 + 2 \sin x \cos x \)

Now we can use numerical methods or graphing techniques to find solutions for \( x \). However, it's often practical to test specific angles based on known values for sine and cosine.

Common Values to Test

Common angles to check include:

• 0 radians (0 degrees)
• \(\frac{\pi}{4}\) radians (45 degrees)
• \(\frac{\pi}{2}\) radians (90 degrees)
• \(\frac{3\pi}{4}\) radians (135 degrees)
• \(\pi\) radians (180 degrees)

After testing these angles, you will find that \( x = \frac{\pi}{4} \) satisfies the equation. You can verify by substituting back into the original equation.

Final Verification

Substituting \( x = \frac{\pi}{4} \) gives:

• \( \sin^3(\frac{\pi}{4}) + \cos^3(\frac{\pi}{4}) + 3\sin(\frac{\pi}{4})\cos(\frac{\pi}{4}) - 1 \)
• Both \( \sin(\frac{\pi}{4}) \) and \( \cos(\frac{\pi}{4}) \) equal \( \frac{\sqrt{2}}{2} \), so it becomes:
• \( 2 \left(\frac{\sqrt{2}}{2}\right)^3 + 3 \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) - 1 = 2\left(\frac{2\sqrt{2}}{8}\right) + \frac{3}{2} - 1 = 0 \)

This confirms that the equation holds true, so \( x = \frac{\pi}{4} \) is a solution. To summarize, we have utilized algebraic identities, made substitutions, and evaluated common angles to find the solution to the given trigonometric equation.