Solving for 2sinθ - cosθ when sinθ 2cosθ 1

Many complex trigonometric expressions can be simplified using fundamental identities and algebraic manipulations. This article dives into solving an intriguing trigonometric problem: if sinθ 2cosθ 1, what would be the value of 2sinθ - cosθ? We will walk through the detailed steps to reach the solution.

1. Transforming the Given Equation

Given that sinθ 2cosθ 1, our first step is to manipulate and simplify this equation to find useful expressions.

Squaring both sides:

(sinθ 2cosθ)2 12

This expands to:

sin2θ 4cos2θ 4sinθcosθ 1

We can use the Pythagorean identity sin2θ cos2θ 1 to further simplify our equation.

2. Applying Trigonometric Identities

From the identity sin2θ cos2θ 1, we subtract it from the expanded equation we derived:

sin2θ 4cos2θ 4sinθcosθ 1

(sin2θ cos2θ) 1

Subtracting these, we get:

3cos2θ 4sinθcosθ 0

This reveals a simplified form we can utilize in our subsequent calculations.

3. Introducing a New Variable S

To find the value of 2sinθ - cosθ, we introduce a new variable, S, where:

S 2sinθ - cosθ

Squaring both sides of this equation:

S2 (2sinθ - cosθ)2

Expanding this, we get:

S2 4sin2θ - 4sinθcosθ cos2θ

Recall the identity sin2θ cos2θ 1, and rearrange to write:

S2 4(1 - cos2θ) - 4sinθcosθ cos2θ

Since we know from our earlier manipulation that:

3cos2θ 4sinθcosθ 0 or 4sinθcosθ -3cos2θ

Substitute 4sinθcosθ -3cos2θ into the squared equation:

S2 4(1 - cos2θ) 3cos2θ

Further simplifying:

S2 4 - cos2θ

Given our earlier simplification, we conclude:

S2 4

Therefore, S ±2 (discarding the negative solution for simplicity).

4. Alternative Method Using Specific Values

Another method involves substituting specific known values for sinθ and cosθ. Assume:

sinβ 1/√5, then cosβ 2/√5

Given the expression sinθ 2cosθ 1, substitute sinθ 1/√5 and cosθ 2/√5:

sinθ 2cosθ (1/√5) 2(2/√5) (1 4)/√5 5/√5 √5

This implies:

sin(θ - β) 2/√5

Leading to:

2sinθ - cosθ 2

Hence, we have a complete set of solutions for 2sinθ - cosθ under the given conditions.

Conclusion

Through careful manipulation of trigonometric identities and substitutions, we have determined the value of 2sinθ - cosθ. The solution reaffirms the versatility and power of trigonometry in resolving complex expressions. This method not only provides a solution but also introduces valuable techniques for handling similar equations in the future.