Question

find the sine, cosine, and tangent of ∠g. simplify your answers and write them as proper fractions, improper fractions, or whole numbers.

for the right triangle with right angle at h, side hi = 34, side hi = 30 (wait, maybe side hi is hypotenuse? wait, the triangle has vertices h (right angle), i, g. side hi is 30, hypotenuse ig is 34? wait, need to find the missing side first. then find sin(g), cos(g), tan(g).

Explanation:

Step1: Encontrar el lado faltante (GH)

Usamos el Teorema de Pitágoras en el triángulo rectángulo \( \triangle GHI \) (con \( \angle H = 90^\circ \)): \( GI^2 = HI^2 + GH^2 \). Sabemos que \( GI = 34 \) y \( HI = 30 \). Entonces:
$$34^2 = 30^2 + GH^2$$
$$1156 = 900 + GH^2$$
$$GH^2 = 1156 - 900 = 256$$
$$GH = \sqrt{256} = 16$$

Step2: Definir los lados para \( \angle G \)

• Lado opuesto (opp) a \( \angle G \): \( HI = 30 \)
• Lado adyacente (adj) a \( \angle G \): \( GH = 16 \)
• Hipotenusa (hyp): \( GI = 34 \)

Step3: Calcular \( \sin(G) \)

La fórmula para el seno es \( \sin(\theta) = \frac{\text{opp}}{\text{hyp}} \). Entonces:
$$\sin(G) = \frac{HI}{GI} = \frac{30}{34} = \frac{15}{17}$$

Step4: Calcular \( \cos(G) \)

La fórmula para el coseno es \( \cos(\theta) = \frac{\text{adj}}{\text{hyp}} \). Entonces:
$$\cos(G) = \frac{GH}{GI} = \frac{16}{34} = \frac{8}{17}$$

Step5: Calcular \( \tan(G) \)

La fórmula para la tangente es \( \tan(\theta) = \frac{\text{opp}}{\text{adj}} \). Entonces:
$$\tan(G) = \frac{HI}{GH} = \frac{30}{16} = \frac{15}{8}$$

Answer:

• \( \sin(G) = \boldsymbol{\frac{15}{17}} \)
• \( \cos(G) = \boldsymbol{\frac{8}{17}} \)
• \( \tan(G) = \boldsymbol{\frac{15}{8}} \)