Solving Trigonometric Equations: sinθcosθ √2

Trigonometric equations are an essential part of mathematics, particularly in trigonometry. This article focuses on solving the equation sinθcosθ √2space; and exploring the various methods to find the value of θ. By the end of this discussion, you will understand the different approaches to solving such equations and the importance of checking for extraneous solutions.

Introduction to the Trigonometric Equation

The problem at hand is the trigonometric equation: sinθcosθ √2. This equation combines both sine and cosine functions, making it a bit more complex than standard trigonometric equations. However, with some algebraic manipulation and trigonometric identities, we can solve for θ.

Solving sinθcosθ √2

Method 1: Using the Identity for sin(2θ)

To solve sinθcosθ √2, we first use the double angle identity:

sinθcosθ 1/2 middot; 2sinθcosθ 1/2 middot; sin2θ

Substituting this into the original equation:

1/2 middot; sin2θ √2

Multiplying both sides by 2:

sin2θ 2√2

However, the maximum value of sinξ is 1, and 2√2 is greater than 1. Therefore, there is no solution in this case. This indicates that our initial assumption was incorrect, and we need to use a different approach.

Method 2: Using Angle Addition Formulas

Another approach is to use the angle addition formula. Recall that:

sin(θ π/4) cos(θ)√2/2 sin(θ)√2/2

Multiplying through by 2:

2sin(θ π/4) cos(θ) sin(θ) √2

Simplifying further:

cos(θ) sin(θ) √2

Since sin(θ) cos(π/4) and cos(θ) sin(π/4), we can rewrite the equation as:

sin(θ) cos(θ) √2

The maximum value of sin(θ) cos(θ) is √2, which occurs when θ π/4. Thus:

θ π/4 2kπ

Method 3: Squaring Both Sides

Another method is to square both sides of the original equation:

(sinθcosθ)^2 (√2)^2

Simplifying:

sin^2θcos^2θ 2

Using the identity sin^2θ cos^2θ 1:

sin^2θ(1 - sin^2θ) 2

Letting u sinθ and v cosθ, we get:

uv √2

Since u^2 v^2 1, we can solve for u and v.

Conclusion

In conclusion, we explored different methods to solve the trigonometric equation sinθcosθ √2. The solution to this equation is θ π/4 2kπ, where k is an integer. Squaring both sides led to extraneous roots that must be checked. It is crucial to verify any solution obtained from squaring both sides to ensure its validity.

Understanding these techniques is vital in solving more complex trigonometric equations and is particularly useful in various fields such as physics, engineering, and mathematics.