In right triangle PQR, the right angle is at vertex Q. If the side lengths are PQ = 15 and… — ACT

In right triangle $PQR$ , the right angle is at vertex $Q$ . If the side lengths are $PQ = 15$ and $QR = 8$ , what is the value of $\tan(P)$ ?

The tangent of angle

P

\frac{8}{15}

The tangent of angle

P

is the ratio of the opposite side (

QR = 8

) to the adjacent side (

PQ = 15

). This gives the value

\frac{8}{15}

Identify the reference angle and the sides of the right triangle relative to it.

The reference angle is

P

. The side opposite to angle

P

QR

with a length of

8

. The side adjacent to angle

P

PQ

with a length of

15

. The hypotenuse is

PR

To calculate a trigonometric ratio, we must first determine which sides are opposite, adjacent, and the hypotenuse relative to the target angle.

Recall the definition of the tangent ratio in a right triangle.

The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side:

\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}

The question asks for the tangent of angle

P

Substitute the identified side lengths into the tangent formula.

\tan(P) = \frac{QR}{PQ} = \frac{8}{15}

Plugging the lengths of the opposite side (

8

) and the adjacent side (

15

) into the tangent ratio yields the final value.

Right triangle trigonometry ratios (SOHCAHTOA), specifically the tangent ratio definition.

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