Question

what are the correct trigonometric ratios for angle h in a right triangle if gi = 5, hi = 12, and gh = 13?
sin h = \frac{12}{13}, cos h = \frac{5}{13}, tan h = \frac{5}{12}
sin h = \frac{5}{13}, cos h = \frac{12}{13}, tan h = \frac{5}{12}
sin h = \frac{12}{13}, cos h = \frac{5}{13}, tan h = \frac{12}{5}
sin h = \frac{5}{13}, cos h = \frac{12}{13}, tan h = \frac{12}{5}

Explanation:

Step1: Recall trigonometric - ratio definitions

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$ for an angle $\theta$.
For angle $H$ in right - triangle $GIH$, the opposite side to angle $H$ is $GI = 5$, the adjacent side to angle $H$ is $HI = 12$, and the hypotenuse is $GH = 13$.

Step2: Calculate $\sin H$

$\sin H=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{GI}{GH}=\frac{5}{13}$

Step3: Calculate $\cos H$

$\cos H=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{HI}{GH}=\frac{12}{13}$

Step4: Calculate $\tan H$

$\tan H=\frac{\text{opposite}}{\text{adjacent}}=\frac{GI}{HI}=\frac{5}{12}$

Answer:

$\sin H=\frac{5}{13},\cos H=\frac{12}{13},\tan H=\frac{5}{12}$ (the second option in the multiple - choice list)