If \theta is an angle in Quadrant IV such that \cos\theta = \frac{4}{5}, what is the value… — ACT

If $\theta$ is an angle in Quadrant IV such that $\cos\theta = \frac{4}{5}$ , what is the value of $\frac{\tan\theta + \sec\theta}{\csc\theta}$ ?

-\frac{3}{10}

Using the Pythagorean identity

\sin^2\theta + \cos^2\theta = 1

in Quadrant IV yields

\sin\theta = -\frac{3}{5}

. Substituting this into quotient and reciprocal identities gives

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

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

, and

\csc\theta = -\frac{5}{3}

. Evaluating

\frac{\tan\theta + \sec\theta}{\csc\theta}

gives

\frac{1/2}{-5/3} = -\frac{3}{10}

Determine

\sin\theta

using the Pythagorean identity

\sin^2\theta + \cos^2\theta = 1

and quadrant sign rules.

\sin\theta = -\sqrt{1 - \left(\frac{4}{5}\right)^2} = -\sqrt{\frac{9}{25}} = -\frac{3}{5}

because sine is negative in Quadrant IV.

The Pythagorean identity relates sine and cosine, and the angle's quadrant determines the sign of the trigonometric ratio.

Calculate the values of

\tan\theta

\sec\theta

, and

\csc\theta

using reciprocal and quotient identities.

\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{-3/5}{4/5} = -\frac{3}{4}

\sec\theta = \frac{1}{\cos\theta} = \frac{5}{4}

, and

\csc\theta = \frac{1}{\sin\theta} = -\frac{5}{3}

Fundamental quotient and reciprocal identities define these functions in terms of sine and cosine.

Substitute the trigonometric values into the given expression

\frac{\tan\theta + \sec\theta}{\csc\theta}

and simplify.

\frac{-\frac{3}{4} + \frac{5}{4}}{-\frac{5}{3}} = \frac{\frac{2}{4}}{-\frac{5}{3}} = \frac{\frac{1}{2}}{-\frac{5}{3}} = \frac{1}{2} \times \left(-\frac{3}{5}\right) = -\frac{3}{10}

Combining terms in the numerator and dividing by the fraction in the denominator yields the simplified value.

Pythagorean, quotient, and reciprocal trigonometric identities with quadrant-dependent signs

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