Given that \theta is an acute angle such that \tan \theta = \frac{4}{3}, calculate the num… — JAMB UTME

Given that $\theta$ is an acute angle such that $\tan \theta = \frac{4}{3}$ , calculate the numerical value of the expression $\frac{3\sin \theta + 2\cos \theta}{3\sin \theta - \cos \theta}$ .

Cevap: 2

Cevap

The exact numerical value of the given expression is 2.

Dividing both the numerator and denominator of

\frac{3\sin \theta + 2\cos \theta}{3\sin \theta - \cos \theta}

\cos \theta

gives

\frac{3\tan \theta + 2}{3\tan \theta - 1}

. Substituting

\tan \theta = \frac{4}{3}

yields

\frac{3(4/3) + 2}{3(4/3) - 1} = \frac{4 + 2}{4 - 1} = \frac{6}{3} = 2

Adım Adım Çözüm

Express sine and cosine terms in terms of tangent or find individual ratio values

Divide every term in the numerator and denominator by

\cos \theta

to obtain

\frac{3\tan \theta + 2}{3\tan \theta - 1}

. Alternatively, using a right triangle with opposite side = 4 and adjacent side = 3 gives hypotenuse = 5, so

\sin \theta = \frac{4}{5}

and

\cos \theta = \frac{3}{5}

Converting to

\tan \theta

simplifies the calculation directly without evaluating square roots or hypotenuse.

Substitute the value of

\tan \theta = \frac{4}{3}

into the expression

Numerator:

3\left(\frac{4}{3}\right) + 2 = 4 + 2 = 6

. Denominator:

3\left(\frac{4}{3}\right) - 1 = 4 - 1 = 3

Simplifies numerical fractions in both parts of the fraction.

Divide numerator by denominator

\frac{6}{3} = 2.

Yields the final integer solution.

Anahtar Kavram

Basic Trigonometric Ratios and Quotients

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