If 3sin θ + 5cos θ = 5, then the value of 5sin θ - 3cos θ is equal to:
3
Solving Trigonometric Equations: Finding the Value of 5sinθ - 3cosθ
The problem asks us to find the value of a specific trigonometric expression, 5sinθ - 3cosθ, given another trigonometric equation, 3sinθ + 5cosθ = 5.
Understanding the Given Trigonometric Equation
We are given the equation:
\( 3\sin\theta + 5\cos\theta = 5 \)
We need to find the value of:
\( 5\sin\theta - 3\cos\theta \)
Step-by-Step Solution using Trigonometric Identity
Let the expression we need to find be equal to \(x\). So, we have a system of two equations:
- \( 3\sin\theta + 5\cos\theta = 5 \) (Equation 1)
- \( 5\sin\theta - 3\cos\theta = x \) (Equation 2)
This is a common pattern in trigonometry problems. A useful technique is to square both equations and add them. This helps eliminate the terms involving \( \sin\theta\cos\theta \) and allows us to use the fundamental trigonometric identity \( \sin^2\theta + \cos^2\theta = 1 \).
Step 1: Square Equation 1
Square both sides of Equation 1:
\( (3\sin\theta + 5\cos\theta)^2 = 5^2 \)
Expand the left side:
\( (3\sin\theta)^2 + (5\cos\theta)^2 + 2(3\sin\theta)(5\cos\theta) = 25 \)
\( 9\sin^2\theta + 25\cos^2\theta + 30\sin\theta\cos\theta = 25 \) (Equation 3)
Step 2: Square Equation 2
Square both sides of Equation 2:
\( (5\sin\theta - 3\cos\theta)^2 = x^2 \)
Expand the left side:
\( (5\sin\theta)^2 + (-3\cos\theta)^2 + 2(5\sin\theta)(-3\cos\theta) = x^2 \)
\( 25\sin^2\theta + 9\cos^2\theta - 30\sin\theta\cos\theta = x^2 \) (Equation 4)
Step 3: Add Equation 3 and Equation 4
Now, add the results from Step 1 and Step 2 (Equation 3 and Equation 4):
\( (9\sin^2\theta + 25\cos^2\theta + 30\sin\theta\cos\theta) + (25\sin^2\theta + 9\cos^2\theta - 30\sin\theta\cos\theta) = 25 + x^2 \)
Combine the like terms:
\( (9\sin^2\theta + 25\sin^2\theta) + (25\cos^2\theta + 9\cos^2\theta) + (30\sin\theta\cos\theta - 30\sin\theta\cos\theta) = 25 + x^2 \)
\( 34\sin^2\theta + 34\cos^2\theta + 0 = 25 + x^2 \)
Step 4: Use the Pythagorean Identity
Factor out 34 from the terms on the left side:
\( 34(\sin^2\theta + \cos^2\theta) = 25 + x^2 \)
Using the fundamental trigonometric identity \( \sin^2\theta + \cos^2\theta = 1 \):
\( 34(1) = 25 + x^2 \)
\( 34 = 25 + x^2 \)
Step 5: Solve for \( x^2 \)
Rearrange the equation to solve for \( x^2 \):
\( x^2 = 34 - 25 \)
\( x^2 = 9 \)
Step 6: Solve for \( x \)
Take the square root of both sides to find \( x \):
\( x = \pm\sqrt{9} \)
\( x = \pm 3 \)
So, the possible values for \( 5\sin\theta - 3\cos\theta \) are 3 and -3.
Analyzing the Possible Values
We found that the value of \( 5\sin\theta - 3\cos\theta \) can be either 3 or -3. The options provided are 3, 4, None of these, and 5. Since 3 is one of the options, and it is a mathematically valid result derived from the given equation, it is the intended answer. The existence of an angle \( \theta \) satisfying \( 3\sin\theta + 5\cos\theta = 5 \) is guaranteed because \( 5^2 \le 3^2 + 5^2 \) (i.e., \( 25 \le 9 + 25 = 34 \)). We also showed during the thought process that specific values of \( \sin\theta \) and \( \cos\theta \) exist for which \( 3\sin\theta + 5\cos\theta = 5 \) and \( 5\sin\theta - 3\cos\theta = 3 \).
Therefore, based on the options provided, the value is 3.
Revision Table: Key Concepts Used
| Concept | Description | Formula/Identity |
|---|---|---|
| Squaring Binomials | Expanding expressions of the form \((a+b)^2\) and \((a-b)^2\). | \((a+b)^2 = a^2 + b^2 + 2ab\) \((a-b)^2 = a^2 + b^2 - 2ab\) |
| Pythagorean Identity | A fundamental identity relating sine and cosine. | \( \sin^2\theta + \cos^2\theta = 1 \) |
| Solving Algebraic Equations | Isolating the unknown variable after simplification. | \( x^2 = k \implies x = \pm\sqrt{k} \) |
Additional Information on Trigonometric Expressions
Expressions of the form \( a\sin\theta + b\cos\theta \) can be rewritten as \( R\sin(\theta + \alpha) \) or \( R\cos(\theta - \beta) \), where \( R = \sqrt{a^2 + b^2} \). The maximum value of \( a\sin\theta + b\cos\theta \) is \( \sqrt{a^2 + b^2} \) and the minimum value is \( -\sqrt{a^2 + b^2} \).
For an equation \( a\sin\theta + b\cos\theta = c \) to have a solution, it must be true that \( |c| \le \sqrt{a^2 + b^2} \), or \( c^2 \le a^2 + b^2 \). In our problem, \( a=3 \), \( b=5 \), \( c=5 \). We have \( 5^2 = 25 \) and \( 3^2 + 5^2 = 9 + 25 = 34 \). Since \( 25 \le 34 \), the given equation \( 3\sin\theta + 5\cos\theta = 5 \) has real solutions for \( \theta \).
The problem structure \( a\sin\theta + b\cos\theta = c \) and finding \( b\sin\theta - a\cos\theta = x \) always leads to \( x^2 = a^2 + b^2 - c^2 \), giving \( x = \pm\sqrt{a^2 + b^2 - c^2} \).
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