Solving Sin Θ + Cos Θ = √2 A Comprehensive Guide

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In this comprehensive guide, we will delve into solving the trigonometric equation sin θ + cos θ = √2. This equation is a classic example that demonstrates various techniques in trigonometry, and understanding its solution provides a solid foundation for tackling more complex problems. We will explore different methods to arrive at the solution and discuss the general solution within the interval 0 ≤ θ < 2π. This guide is designed to be both educational and SEO-friendly, ensuring that readers not only grasp the solution but also understand the underlying concepts.

1. Introduction to the Trigonometric Equation

The trigonometric equation sin θ + cos θ = √2 involves two fundamental trigonometric functions: sine and cosine. Solving such equations requires a strategic approach, often involving the use of trigonometric identities and algebraic manipulations. The goal is to find the values of θ that satisfy the equation within a specified interval, which in this case is 0 ≤ θ < 2π. This interval represents a full circle in radians, covering all possible angles.

Trigonometric equations are a cornerstone of mathematics, appearing in various applications ranging from physics to engineering. Understanding how to solve them is crucial for anyone studying these fields. This particular equation, sin θ + cos θ = √2, is a great starting point due to its simplicity and the clear steps involved in its solution. Let's embark on this mathematical journey with a structured approach, ensuring that every step is clear and well-explained.

2. Method 1: Transforming the Equation Using Trigonometric Identities

One of the most elegant methods to solve sin θ + cos θ = √2 involves transforming the equation into a more manageable form using trigonometric identities. Specifically, we can utilize the identity that relates sine and cosine functions with a phase shift. This method is particularly effective because it simplifies the equation into a single trigonometric function, making it easier to solve.

2.1 Multiplying by a Suitable Constant

The first step is to multiply both sides of the equation by a constant that allows us to express the left-hand side in the form of a single trigonometric function. The constant we choose is 1/√2. This might seem arbitrary at first, but it's a strategic choice based on the coefficients of the sine and cosine terms. Multiplying the equation by 1/√2, we get:

(1/√2)sin θ + (1/√2)cos θ = √2 * (1/√2)

Simplifying, we have:

(1/√2)sin θ + (1/√2)cos θ = 1

2.2 Recognizing Trigonometric Values

Now, we recognize that 1/√2 is the value of both sin(π/4) and cos(π/4). This allows us to rewrite the equation in terms of these trigonometric values:

sin(π/4)sin θ + cos(π/4)cos θ = 1

This form is crucial because it aligns with the cosine addition formula.

2.3 Applying the Cosine Addition Formula

The cosine addition formula states that:

cos(A - B) = cos A cos B + sin A sin B

Comparing this with our equation, we can see that it fits perfectly. By applying this formula, we can rewrite the equation as:

cos(θ - π/4) = 1

This transformation is a key step in simplifying the problem. We have successfully converted the original equation into one involving a single trigonometric function, which is much easier to solve.

3. Solving the Simplified Equation

Now that we have the simplified equation cos(θ - π/4) = 1, we can proceed to find the values of θ that satisfy it. This involves understanding the behavior of the cosine function and identifying the angles for which it equals 1.

3.1 Understanding the Cosine Function

The cosine function reaches its maximum value of 1 at integer multiples of 2π. In other words, cos(x) = 1 when x = 2nπ, where n is an integer. This fundamental property of the cosine function is essential for solving our equation.

3.2 Finding the General Solution

Applying this knowledge to our equation, cos(θ - π/4) = 1, we can write:

θ - π/4 = 2nπ, where n is an integer

This is the general solution to the equation, representing all possible values of θ that satisfy it. However, we are interested in solutions within the interval 0 ≤ θ < 2π.

3.3 Finding Solutions Within the Interval 0 ≤ θ < 2π

To find the specific solutions within the interval 0 ≤ θ < 2π, we need to substitute different integer values for n and check if the resulting θ falls within the interval.

  • For n = 0:

    θ - π/4 = 2(0)π

    θ = π/4

    This solution falls within our interval.

  • For n = 1:

    θ - π/4 = 2(1)π

    θ = 2π + π/4

    θ = 9π/4

    This solution is greater than 2π and therefore outside our interval.

  • For n = -1:

    θ - π/4 = 2(-1)π

    θ = -2π + π/4

    θ = -7π/4

    This solution is less than 0 and therefore outside our interval.

Therefore, the only solution within the interval 0 ≤ θ < 2π is θ = π/4.

4. Method 2: Squaring Both Sides of the Equation

Another method to solve sin θ + cos θ = √2 involves squaring both sides of the equation. This technique can introduce extraneous solutions, so it's crucial to verify each solution obtained at the end. However, it provides a different perspective and reinforces the importance of checking solutions in trigonometric equations.

4.1 Squaring Both Sides

Squaring both sides of the equation sin θ + cos θ = √2, we get:

(sin θ + cos θ)² = (√2)²

Expanding the left side, we have:

sin² θ + 2sin θ cos θ + cos² θ = 2

4.2 Using the Pythagorean Identity

We know that sin² θ + cos² θ = 1 (the Pythagorean identity). Substituting this into the equation, we get:

1 + 2sin θ cos θ = 2

4.3 Simplifying the Equation

Subtracting 1 from both sides, we have:

2sin θ cos θ = 1

4.4 Applying the Double Angle Formula

The double angle formula for sine is sin(2θ) = 2sin θ cos θ. Using this, we can rewrite the equation as:

sin(2θ) = 1

This equation is now in a simpler form, involving a single trigonometric function.

5. Solving sin(2θ) = 1

Now, we need to find the values of θ that satisfy the equation sin(2θ) = 1. Similar to the cosine function, we need to understand the behavior of the sine function and identify the angles for which it equals 1.

5.1 Understanding the Sine Function

The sine function reaches its maximum value of 1 at angles of the form (π/2) + 2nπ, where n is an integer. In other words, sin(x) = 1 when x = (π/2) + 2nπ.

5.2 Finding the General Solution for 2θ

Applying this to our equation, sin(2θ) = 1, we can write:

2θ = (π/2) + 2nπ, where n is an integer

5.3 Finding the General Solution for θ

To find the general solution for θ, we divide both sides by 2:

θ = (π/4) + nπ, where n is an integer

5.4 Finding Solutions Within the Interval 0 ≤ θ < 2π

Now we need to find the specific solutions within the interval 0 ≤ θ < 2π by substituting different integer values for n:

  • For n = 0:

    θ = (π/4) + 0π

    θ = π/4

    This solution falls within our interval.

  • For n = 1:

    θ = (π/4) + 1π

    θ = 5π/4

    This solution also falls within our interval.

  • For n = 2:

    θ = (π/4) + 2π

    θ = 9π/4

    This solution is greater than 2π and therefore outside our interval.

  • For n = -1:

    θ = (π/4) - π

    θ = -3π/4

    This solution is less than 0 and therefore outside our interval.

So, we have two potential solutions: θ = π/4 and θ = 5π/4.

6. Verifying the Solutions

Since we squared both sides of the equation, it's crucial to verify each potential solution to ensure they are not extraneous.

6.1 Verifying θ = π/4

Substitute θ = π/4 into the original equation:

sin(π/4) + cos(π/4) = √2

(1/√2) + (1/√2) = √2

2/√2 = √2

√2 = √2

This solution is valid.

6.2 Verifying θ = 5π/4

Substitute θ = 5π/4 into the original equation:

sin(5π/4) + cos(5π/4) = √2

(-1/√2) + (-1/√2) = √2

-2/√2 = √2

-√2 = √2

This solution is extraneous.

Therefore, the only valid solution within the interval 0 ≤ θ < 2π is θ = π/4.

7. Conclusion

In this comprehensive guide, we explored two different methods to solve the trigonometric equation sin θ + cos θ = √2. The first method involved transforming the equation using trigonometric identities, specifically the cosine addition formula. This method led us directly to the solution θ = π/4. The second method involved squaring both sides of the equation, which introduced a potential extraneous solution. After verification, we confirmed that θ = π/4 is the only valid solution within the interval 0 ≤ θ < 2π.

Understanding trigonometric equations is crucial for various applications in mathematics, physics, and engineering. The techniques discussed in this guide provide a solid foundation for tackling more complex problems. Remember, when solving trigonometric equations, it's always a good practice to verify the solutions, especially when squaring both sides, to avoid extraneous solutions. Mastering these techniques will enhance your problem-solving skills and deepen your understanding of trigonometry.

In summary, the solution to the equation sin θ + cos θ = √2 within the interval 0 ≤ θ < 2π is θ = π/4. This example illustrates the power of trigonometric identities and the importance of careful verification in solving trigonometric equations.