Evaluate Trigonometric Expression
The problem asks for the value of the expression $\frac{\cos \theta - \tan \theta}{2 \cot \theta}$ given that $\sin \theta = \frac{5}{13}$ and $\theta$ is a positive acute angle.
Calculate Trigonometric Values
First, find $\cos \theta$. Using the identity $\sin^2 \theta + \cos^2 \theta = 1$, and knowing $\theta$ is acute (so $\cos \theta > 0$):
$ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{169 - 25}{169} = \frac{144}{169} $
$ \cos \theta = \sqrt{\frac{144}{169}} = \frac{12}{13} $
Next, find $\tan \theta$ and $\cot \theta$:
$ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{5/13}{12/13} = \frac{5}{12} $
$ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{5/12} = \frac{12}{5} $
Evaluate the Expression
Substitute the calculated values into the given expression:
$ \frac{\cos \theta - \tan \theta}{2 \cot \theta} = \frac{\frac{12}{13} - \frac{5}{12}}{2 \times \frac{12}{5}} $
Calculate the numerator:
$ \frac{12}{13} - \frac{5}{12} = \frac{12 \times 12 - 5 \times 13}{13 \times 12} = \frac{144 - 65}{156} = \frac{79}{156} $
Calculate the denominator:
$ 2 \times \frac{12}{5} = \frac{24}{5} $
Divide the numerator by the denominator:
$ \frac{79/156}{24/5} = \frac{79}{156} \times \frac{5}{24} = \frac{79 \times 5}{156 \times 24} = \frac{395}{3744} $
The value of the expression is $\frac{395}{3744}$.
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 \(\frac{3 \sin 58^{\circ}}{\cos 32^{\circ}}+\frac{3 \sin 42^{\circ}}{\cos 48^{\circ}}\) ?
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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 ?
