In right triangle PQR, the measure of angle Q is 90^\circ. If \tan(P) = \frac{4}{3}, what … — SAT

In right triangle $PQR$ , the measure of angle $Q$ is $90^\circ$ . If $\tan(P) = \frac{4}{3}$ , what is the value of $\sin(P)$ ?

\frac{4}{5}

The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. Given that

\tan(P) = \frac{4}{3}

, we can set the length of the side opposite to angle P as

4

and the adjacent side as

3

. Using the Pythagorean theorem, the hypotenuse is

\sqrt{3^2 + 4^2} = 5

. Since the sine of an angle is the ratio of the opposite side to the hypotenuse,

\sin(P) = \frac{4}{5}

Identify the relationship between the tangent ratio and the sides of the right triangle.

The tangent of angle P is defined as the opposite side divided by the adjacent side. Since

\tan(P) = \frac{4}{3}

, we can set the opposite side to

4

and the adjacent side to

3

This allows us to establish the relative lengths of the sides of the right triangle.

Calculate the length of the hypotenuse using the Pythagorean theorem.

The hypotenuse is

\sqrt{3^2 + 4^2} = \sqrt{25} = 5

The hypotenuse is required to calculate the sine ratio.

Determine the sine of angle P.

The sine of angle P is defined as the opposite side divided by the hypotenuse, which is

\frac{4}{5}

This yields the final requested trigonometric ratio.

Using basic trigonometric ratios (SOH CAH TOA) and the Pythagorean theorem to evaluate trigonometric ratios in a right triangle.

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