Question

note: triangle is not to scale.
in
ight\ triangle abc above, what is the sine of a?
options:
\\(\frac{6}{7}\\), \\(\frac{9}{6}\\),
\\(\frac{6}{9}\\), \\(\frac{7}{6}\\),
\\(\frac{9}{7}\\), \\(\frac{7}{9}\\)
question help: video

Explanation:

Step1: Recall Sine Definition

In a right triangle, $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$.

Step2: Identify Sides for Angle A

For $\angle A$, opposite side is $BC = 7$, hypotenuse is $AB = 9$.

Step3: Calculate $\sin(A)$

$\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{7}{9}$? Wait, no—wait, angle A: let's recheck. Wait, right angle at C, so triangle ABC: right-angled at C. So sides: AC = 6, BC = 7, AB = 9 (hypotenuse). For angle A: opposite side is BC (length 7), adjacent is AC (6), hypotenuse AB (9). Wait, no: in right triangle, angle A: the sides: opposite to A is BC (since angle A is at vertex A, so the side opposite is BC, which is 7), hypotenuse is AB (9). Wait, but wait, let's confirm: in triangle ABC, right-angled at C. So vertices: C (right angle), A and B. So side AC: length 6 (from A to C), BC: length 7 (from B to C), AB: length 9 (hypotenuse, from A to B). So angle at A: the sides: opposite to angle A is BC (7), adjacent is AC (6), hypotenuse AB (9). So $\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{7}{9}$. Wait, but let's check the options. The options include $\frac{7}{9}$. Wait, let's re-express:

Wait, maybe I mixed up. Let's label the triangle: right angle at C. So sides:

• AC: 6 (vertical leg, from A to C)
• BC: 7 (horizontal leg, from B to C)
• AB: 9 (hypotenuse, from A to B)

Angle at A: so the angle between AC (6) and AB (9). So the side opposite angle A is BC (7), because angle A is at A, so the side opposite is the one not connected to A, which is BC (length 7). Hypotenuse is AB (9). So $\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{7}{9}$. Looking at the options, one of them is $\frac{7}{9}$. So that's the answer.

Wait, let's confirm the sine formula again: $\sin(\theta) = \frac{\text{length of opposite side}}{\text{length of hypotenuse}}$. For angle A, opposite side is BC (7), hypotenuse is AB (9). So $\sin(A) = \frac{7}{9}$.

Answer:

$\frac{7}{9}$ (corresponding to the option with $\frac{7}{9}$)