Solve For { X $}$ In The Equation:${ \sin X + \cos X = \frac{1 + \sqrt{3}}{2} }$

Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of trigonometric functions and their properties. In this article, we will focus on solving the equation $\sin x + \cos x = \frac{1 + \sqrt{3}}{2}$, which involves the sum of sine and cosine functions. We will break down the solution into manageable steps, using various trigonometric identities and properties to simplify the equation and find the value of x.

Understanding the Equation

The given equation is $\sin x + \cos x = \frac{1 + \sqrt{3}}{2}$. To solve this equation, we need to isolate the variable x. However, the equation involves the sum of sine and cosine functions, which makes it challenging to solve directly. We will use various trigonometric identities and properties to simplify the equation and find the value of x.

Using Trigonometric Identities

One of the most useful trigonometric identities is the Pythagorean identity, which states that $\sin^2 x + \cos^2 x = 1$. We can use this identity to simplify the equation. However, in this case, we have the sum of sine and cosine functions, not their squares. Therefore, we need to use a different approach.

Squaring Both Sides

One way to simplify the equation is to square both sides. This will allow us to use the Pythagorean identity to eliminate the square roots. Squaring both sides of the equation gives us:

$$\sin^2 x + 2 \sin x \cos x + \cos^2 x = \left(\frac{1 + \sqrt{3}}{2}\right)^2$$

Using the Pythagorean Identity

Now we can use the Pythagorean identity to simplify the equation. We know that $\sin^2 x + \cos^2 x = 1$, so we can substitute this into the equation:

$$1 + 2 \sin x \cos x = \left(\frac{1 + \sqrt{3}}{2}\right)^2$$

Simplifying the Equation

Now we can simplify the equation by expanding the right-hand side:

$$1 + 2 \sin x \cos x = \frac{1 + 2 \sqrt{3} + 3}{4}$$

$$1 + 2 \sin x \cos x = \frac{4 + 2 \sqrt{3}}{4}$$

Using the Double Angle Formula

We can use the double angle formula to simplify the equation further. The double angle formula states that $\sin 2x = 2 \sin x \cos x$. We can use this formula to rewrite the equation:

$$1 + \sin 2x = \frac{4 + 2 \sqrt{3}}{4}$$

Solving for x

Now we can solve for x by isolating the variable. We can start by subtracting 1 from both sides:

$$\sin 2x = \frac{4 + 2 \sqrt{3}}{4} - 1$$

$$\sin 2x = \frac{2 + 2 \sqrt{3}}{4}$$

Finding the Value of x

Now we can find the value of x by using the inverse sine function. We know that $\sin 2x = \frac{2 + 2 \sqrt{3}}{4}$, so we can take the inverse sine of both sides:

$$2x = \sin^{-1} \left(\frac{2 + 2 \sqrt{3}}{4}\right)$$

Simplifying the Inverse Sine

We can simplify the inverse sine by using the fact that $\sin^{-1} x = \frac{\pi}{2} - \cos^{-1} x$. We can use this formula to rewrite the equation:

$$2x = \frac{\pi}{2} - \cos^{-1} \left(\frac{2 + 2 \sqrt{3}}{4}\right)$$

Finding the Value of x

Now we can find the value of x by dividing both sides by 2:

$$x = \frac{1}{2} \left(\frac{\pi}{2} - \cos^{-1} \left(\frac{2 + 2 \sqrt{3}}{4}\right)\right)$$

Conclusion

In this article, we solved the equation $\sin x + \cos x = \frac{1 + \sqrt{3}}{2}$ using various trigonometric identities and properties. We started by squaring both sides of the equation and using the Pythagorean identity to simplify the equation. We then used the double angle formula to rewrite the equation and solve for x. Finally, we found the value of x by using the inverse sine function and simplifying the result. The final answer is:

$$x = \frac{1}{2} \left(\frac{\pi}{2} - \cos^{-1} \left(\frac{2 + 2 \sqrt{3}}{4}\right)\right)$$

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Trigonometric Equations" by Wolfram MathWorld

Additional Resources

• [1] Khan Academy: Trigonometry
• [2] MIT OpenCourseWare: Trigonometry
• [3] Wolfram Alpha: Trigonometric Equations
  Solving Trigonometric Equations: A Q&A Guide =====================================================

Introduction

In our previous article, we solved the equation $\sin x + \cos x = \frac{1 + \sqrt{3}}{2}$ using various trigonometric identities and properties. However, we know that solving trigonometric equations can be a challenging task, and many students struggle to understand the concepts and techniques involved. In this article, we will provide a Q&A guide to help students better understand the concepts and techniques involved in solving trigonometric equations.

Q: What is a trigonometric equation?

A: A trigonometric equation is an equation that involves trigonometric functions, such as sine, cosine, and tangent. These equations can be used to model real-world problems, such as the motion of objects, the behavior of electrical circuits, and the properties of waves.

Q: What are the different types of trigonometric equations?

A: There are several types of trigonometric equations, including:

• Linear trigonometric equations: These equations involve a single trigonometric function, such as $\sin x = \frac{1}{2}$.
• Quadratic trigonometric equations: These equations involve the product of two trigonometric functions, such as $\sin^2 x + \cos^2 x = 1$.
• Trigonometric equations with multiple angles: These equations involve the sum or difference of two or more angles, such as $\sin (x + y) = \sin x \cos y + \cos x \sin y$.

Q: How do I solve a trigonometric equation?

A: Solving a trigonometric equation involves several steps, including:

• Simplifying the equation: This involves using trigonometric identities to simplify the equation and make it easier to solve.
• Using algebraic techniques: This involves using algebraic techniques, such as factoring and solving quadratic equations, to solve the equation.
• Using trigonometric properties: This involves using properties of trigonometric functions, such as the Pythagorean identity, to solve the equation.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• Pythagorean identity: $\sin^2 x + \cos^2 x = 1$
• Double angle formula: $\sin 2x = 2 \sin x \cos x$
• Half angle formula: $\sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}}$

Q: How do I use trigonometric identities to solve an equation?

A: To use trigonometric identities to solve an equation, you need to:

• Identify the trigonometric function: Identify the trigonometric function involved in the equation.
• Use the appropriate identity: Use the appropriate trigonometric identity to simplify the equation.
• Solve the resulting equation: Solve the resulting equation using algebraic techniques.

Q: What are some common mistakes to avoid when solving trigonometric equations?

A: Some common mistakes to avoid when solving trigonometric equations include:

• Not simplifying the equation: Failing to simplify the equation can make it difficult to solve.
• Not using the correct trigonometric identity: Using the wrong trigonometric identity can lead to incorrect solutions.
• Not checking the solutions: Failing to check the solutions can lead to incorrect answers.

Conclusion

Solving trigonometric equations can be a challenging task, but with practice and patience, you can become proficient in solving these equations. By understanding the concepts and techniques involved, you can solve a wide range of trigonometric equations and apply them to real-world problems. Remember to simplify the equation, use algebraic techniques, and use trigonometric properties to solve the equation.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Trigonometric Equations" by Wolfram MathWorld

Additional Resources

• [1] Khan Academy: Trigonometry
• [2] MIT OpenCourseWare: Trigonometry
• [3] Wolfram Alpha: Trigonometric Equations