What is the exact value of \csc\left(\arctan\left(-\frac{7}{24}\right)\right)? — ACT

What is the exact value of $\csc\left(\arctan\left(-\frac{7}{24}\right)\right)$ ?

-\frac{25}{7}

Let

\theta = \arctan\left(-\frac{7}{24}\right)

. By definition of the inverse tangent function's principal range

\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

, a negative input produces an angle in Quadrant IV. In Quadrant IV, the opposite side is

-7

, the adjacent side is

24

, and the hypotenuse is

\sqrt{24^2 + (-7)^2} = 25

. Cosecant is the ratio of hypotenuse to opposite, giving

\frac{25}{-7} = -\frac{25}{7}

Determine the quadrant for

\theta = \arctan\left(-\frac{7}{24}\right)

\theta

lies in Quadrant IV because the principal range of

\arctan(x)

\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

and the input is negative.

Inverse tangent maps negative real numbers to angles in the interval

\left(-\frac{\pi}{2}, 0\right)

Set up a right triangle ratio for

\tan(\theta) = -\frac{7}{24}

Opposite side =

-7

, adjacent side =

24

, hypotenuse =

\sqrt{24^2 + (-7)^2} = 25

Tangent is the ratio of opposite to adjacent sides, and the hypotenuse is determined using the Pythagorean theorem.

Calculate

\csc(\theta)

\csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{25}{-7} = -\frac{25}{7}

Cosecant is defined as the reciprocal of sine, which equals hypotenuse divided by opposite side.

Evaluating trigonometric compositions involving inverse trigonometric functions using right triangle geometry and principal angle ranges.

Question