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Difficulty: MediumBasic Trigonometric Ratios, Special Angles, and Identities

If θ\theta is an acute angle such that tan⁡θ=2\tan \theta = 2, calculate the exact numerical value of 3sin⁡θ+cos⁡θsin⁡θ+2cos⁡θ\frac{3\sin \theta + \cos \theta}{\sin \theta + 2\cos \theta}.

Answer: 1.75

Answer

The exact numerical value of the trigonometric expression is 1.75.
Dividing every term in the expression by cos⁡θ\cos \theta converts sin⁡θ\sin \theta terms into tan⁡θ\tan \theta. The quotient becomes 3tan⁡θ+1tan⁡θ+2\frac{3\tan \theta + 1}{\tan \theta + 2}. Substituting tan⁡θ=2\tan \theta = 2 gives 3(2)+12+2=74=1.75\frac{3(2) + 1}{2 + 2} = \frac{7}{4} = 1.75.

Step-by-Step Solution

1
Divide numerator and denominator by cos⁡θ\cos \theta
The expression becomes 3(sin⁡θcos⁡θ)+1sin⁡θcos⁡θ+2=3tan⁡θ+1tan⁡θ+2\frac{3\left(\frac{\sin \theta}{\cos \theta}\right) + 1}{\frac{\sin \theta}{\cos \theta} + 2} = \frac{3\tan \theta + 1}{\tan \theta + 2}.
Using the trigonometric identity tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta} simplifies the calculation without needing to evaluate the hypotenuse.
2
Substitute tan⁡θ=2\tan \theta = 2
3(2)+12+2=74\frac{3(2) + 1}{2 + 2} = \frac{7}{4}
Replace tan⁡θ\tan \theta with its given numerical value of 2.
3
Convert fraction to decimal
1.75
Decimal representation of the exact fraction 74\frac{7}{4}.

Key Concept

Basic Trigonometric Ratios and Quotient Identity
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Topics

Basic Trigonometric Ratios, Special Angles, and Identities

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