If \(\frac{\sin\theta}{x} = \frac{\cos\theta}{y}\), then \(\sin\theta - \cos\theta\) is:
\(\frac{x-y}{\sqrt{x^2+y^2}}\)
We are given the equation:
\(\frac{\sin\theta}{x} = \frac{\cos\theta}{y}\)
We need to determine the value of \(\sin\theta - \cos\theta\).
- First, cross-multiply to remove the fractions from the given equation:
- Rearrange the equation to express one trigonometric function in terms of the other:
- Substitute the expression for \(\sin\theta\) into \(\sin\theta - \cos\theta\):
- Introduce a factor to compare with the root of the square sum:
- Using Pythagorean identity, substitute \(\sin^2\theta + \cos^2\theta = \left(\frac{x}{\sqrt{x^2+y^2}}\right)^2 + \left(\frac{y}{\sqrt{x^2+y^2}}\right)^2 = 1\)
- Finally factor and simplify to find within given options:
Therefore, the correct answer is \(\frac{x-y}{\sqrt{x^2+y^2}}\).
Similar Questions
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Important Questions from Trigonometric Ratios and Identities
If cosec θ = 13/12, then sin θ + cos θ - tan θ is equal to:
What is the value of \(\frac{3 \sin 58^{\circ}}{\cos 32^{\circ}}+\frac{3 \sin 42^{\circ}}{\cos 48^{\circ}}\) ?
If \(\frac{{\tan \theta + \sin \theta }}{{\tan \theta - \sin \theta }} = \frac{{k + 1}}{{k - 1}},\) then k = ?
If α + β = 90° and α = 2β, then the value of 3 cos 2 α - 2 sin 2 β is equal to:
If \(\sqrt{3}\) tan θ = 3 sin θ, then what is the value of sin 2θ − cos 2θ ?
