Question

find the measure of side a.

c = 19 m

(round the answer to the nearest whole number.)

a = \square m

Explanation:

Step1: Identify the trigonometric relation

In right triangle \(ABC\) with right angle at \(C\), angle at \(A\) is \(33^\circ\) and hypotenuse \(c = 19\) m. We need to find side \(a\) (opposite to angle \(A\)? Wait, no, wait: Wait, side \(a\) is opposite to angle \(A\)? Wait, no, in standard notation, side \(a\) is opposite angle \(A\), side \(b\) opposite angle \(B\), hypotenuse \(c\). Wait, but angle at \(A\) is \(33^\circ\), right angle at \(C\), so angle at \(B\) is \(90 - 33=57^\circ\)? Wait, no, wait: Wait, side \(a\) is \(BC\), which is opposite angle \(A\). Wait, sine of angle \(A\) is \(\sin(A)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{a}{c}\). Wait, no: Wait, angle \(A\) is \(33^\circ\), so opposite side is \(a\) (BC), hypotenuse is \(c = AB = 19\) m. So \(\sin(33^\circ)=\frac{a}{19}\), so \(a = 19\times\sin(33^\circ)\).

Step2: Calculate the value

First, find \(\sin(33^\circ)\). Using a calculator, \(\sin(33^\circ)\approx0.5446\). Then \(a = 19\times0.5446\approx10.3474\). Round to the nearest whole number, so \(a\approx10\)? Wait, wait, no, wait: Wait, maybe I mixed up. Wait, angle at \(A\) is \(33^\circ\), so side \(a\) is adjacent? Wait, no, let's re-examine the triangle. Points: \(C\) is right angle, so \(AC\) and \(BC\) are legs, \(AB\) is hypotenuse. Angle at \(A\) is \(33^\circ\), so angle between \(AC\) and \(AB\) is \(33^\circ\). So side \(BC\) (which is \(a\)) is opposite angle \(A\), side \(AC\) is adjacent to angle \(A\), hypotenuse \(AB = c = 19\). So \(\sin(A)=\frac{BC}{AB}=\frac{a}{c}\), so \(a = c\times\sin(A)\). So \(\sin(33^\circ)\approx0.5446\), so \(a = 19\times0.5446\approx10.3474\), which rounds to 10? Wait, no, wait, maybe I made a mistake. Wait, maybe angle at \(A\) is \(33^\circ\), so side \(a\) is opposite angle \(A\), so yes, \(\sin(33^\circ)=\frac{a}{19}\), so \(a = 19\sin(33^\circ)\approx19\times0.5446\approx10.347\), which is approximately 10 when rounded to nearest whole number? Wait, but let's check again. Wait, maybe I had the angle wrong. Wait, maybe angle at \(B\) is \(33^\circ\)? No, the diagram shows angle at \(A\) is \(33^\circ\). Wait, maybe I should use cosine? Wait, no, if \(a\) is adjacent to angle \(A\), then \(\cos(A)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{AC}{AB}\), but \(a\) is \(BC\), which is opposite. Wait, maybe the problem is that I misassigned the side. Wait, the label: side \(a\) is \(BC\), angle at \(A\) is \(33^\circ\), so \(\sin(33^\circ)=\frac{BC}{AB}=\frac{a}{19}\), so \(a = 19\sin(33^\circ)\approx19\times0.5446\approx10.35\), which rounds to 10. Wait, but let's calculate more accurately. \(\sin(33^\circ)\) is approximately 0.544639035, so 19*0.544639035 = 19*0.544639035 = 10.348141665, so rounded to nearest whole number is 10. Wait, but maybe I made a mistake in the trigonometric ratio. Wait, another way: angle at \(A\) is \(33^\circ\), so angle at \(B\) is \(57^\circ\). If we consider side \(a\) as opposite angle \(B\), then \(\sin(57^\circ)=\frac{a}{19}\), since angle \(B\) is \(57^\circ\), opposite side is \(a\). \(\sin(57^\circ)\approx0.8387\), so \(a = 19\times0.8387\approx15.935\), which rounds to 16. Wait, this is a contradiction. So where is the mistake?

Wait, let's look at the triangle labels again. The triangle has vertices \(A\), \(B\), \(C\) with right angle at \(C\). So:

- Vertex \(A\): angle \(33^\circ\)
- Vertex \(C\): right angle
- Vertex \(B\): angle \(90 - 33 = 57^\circ\)

Sides:

- \(AB\): hypotenuse \(c = 19\) m
- \(BC\): side \(a\) (opposite angle \(A\))
- \(AC\): side \(b\) (opposite angle \(B\))

So, for angle \(A = 33^\circ\):

\(\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AB} = \frac{a}{c}\)

So \(a = c \times \sin(A) = 19 \times \sin(33^\circ)\)

\(\sin(33^\circ) \approx 0.5446\)

\(19 \times 0.5446 = 10.3474\), which rounds to 10.

But if we consider angle \(B = 57^\circ\), then \(\sin(B) = \frac{AC}{AB}\), no, \(AC\) is side \(b\), \(BC\) is side \(a\). Wait, no: in standard notation, side \(a\) is opposite angle \(A\), side \(b\) opposite angle \(B\), side \(c\) opposite angle \(C\) (which is the hypotenuse, since angle \(C\) is right angle). So angle \(C = 90^\circ\), so side \(c\) is hypotenuse (opposite right angle). So side \(a\) is opposite angle \(A\) (33°), side \(b\) opposite angle \(B\) (57°), side \(c\) opposite angle \(C\) (90°), so \(c = 19\) m.

Therefore, \(\sin(A) = \frac{a}{c}\) => \(a = c \times \sin(A)\)

So \(a = 19 \times \sin(33^\circ) \approx 19 \times 0.5446 \approx 10.35\), so rounded to nearest whole number is 10. Wait, but maybe the problem is that the side \(a\) is adjacent to angle \(A\). Wait, no, in the diagram, side \(a\) is \(BC\), which is horizontal, \(AC\) is vertical, \(AB\) is hypotenuse. Angle at \(A\) is between \(AC\) and \(AB\), so the sides: \(AC\) is adjacent to angle \(A\), \(BC\) is opposite to angle \(A\), \(AB\) is hypotenuse. So yes, \(\sin(A) = \frac{BC}{AB} = \frac{a}{c}\), so \(a = c \times \sin(A)\). So the calculation is correct. So the answer is 10? Wait, but when I calculated using angle \(B\), I got 16, which is wrong because of misassignment. So the correct ratio is \(\sin(33^\circ)\), so \(a \approx 10\). Wait, but let's check with calculator again. Let's compute \(\sin(33^\circ)\):

Using calculator: \(\sin(33^\circ) \approx 0.5446\)

19 * 0.5446 = 10.3474, so rounded to nearest whole number is 10.

Wait, but maybe the problem is that the side \(a\) is adjacent, so \(\cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AC}{AB}\), but \(a\) is \(BC\), which is opposite. So no, the first calculation is correct. So the answer is 10.

Answer:

10