If \tan \theta = 2, what is the value of \frac{3\sin \theta + 4\cos \theta}{5\sin \theta -… — ACT

If $\tan \theta = 2$ , what is the value of $\frac{3\sin \theta + 4\cos \theta}{5\sin \theta - 2\cos \theta}$ ?

Answer: 1.25

The value of the expression is 1.25.

Using the trigonometric quotient identity

\tan \theta = \frac{\sin \theta}{\cos \theta}

, dividing both the numerator and denominator of

\frac{3\sin \theta + 4\cos \theta}{5\sin \theta - 2\cos \theta}

\cos \theta

transforms the fraction into

\frac{3\tan \theta + 4}{5\tan \theta - 2}

. Substituting

\tan \theta = 2

gives

\frac{3(2) + 4}{5(2) - 2} = \frac{10}{8} = 1.25

. Alternatively, constructing a right triangle with opposite side 2 and adjacent side 1 gives a hypotenuse of

\sqrt{5}

, yielding

\sin \theta = \frac{2}{\sqrt{5}}

and

\cos \theta = \frac{1}{\sqrt{5}}

, which produces the exact same ratio of

\frac{10/\sqrt{5}}{8/\sqrt{5}} = 1.25

Divide every term in both the numerator and the denominator by

\cos \theta

The expression becomes

\frac{3\left(\frac{\sin \theta}{\cos \theta}\right) + 4\left(\frac{\cos \theta}{\cos \theta}\right)}{5\left(\frac{\sin \theta}{\cos \theta}\right) - 2\left(\frac{\cos \theta}{\cos \theta}\right)}

Dividing by

\cos \theta

allows us to convert sine-and-cosine terms into tangent terms using the quotient identity.

Apply the quotient identity

\tan \theta = \frac{\sin \theta}{\cos \theta}

The expression simplifies to

\frac{3\tan \theta + 4}{5\tan \theta - 2}

This reduces the trigonometric expression to an algebraic expression containing only

\tan \theta

Substitute

\tan \theta = 2

into the simplified expression and compute the result.

\frac{3(2) + 4}{5(2) - 2} = \frac{6 + 4}{10 - 2} = \frac{10}{8} = 1.25.

Performing basic arithmetic yields the exact numeric answer.

Quotient Identity of Tangent

Question