Question

find rs.
write your answer as an integer or as a decimal rounded to the nearest tenth.
rs =

Explanation:

Step1: Identify the trigonometric ratio

In right triangle \( QSR \), we know angle \( Q = 57^\circ \), the side opposite to angle \( Q \) is \( RS \), and the side adjacent to angle \( Q \) is \( QS=\sqrt{14} \). We use the tangent function, which is defined as \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). So, \( \tan(57^\circ)=\frac{RS}{QS} \).

Step2: Solve for \( RS \)

Substitute \( QS = \sqrt{14} \) and \( \theta = 57^\circ \) into the formula. First, calculate \( \sqrt{14}\approx3.7417 \). Then, \( RS=\tan(57^\circ)\times QS \). We know that \( \tan(57^\circ)\approx1.5399 \). So, \( RS\approx1.5399\times3.7417 \).

Step3: Calculate the value

\( 1.5399\times3.7417\approx5.76 \), and rounded to the nearest tenth is \( 5.8 \). Wait, wait, let's recalculate more accurately. Wait, \( \sqrt{14}\approx3.741657387 \), \( \tan(57^\circ)\approx1.539864964 \). Then \( 3.741657387\times1.539864964\approx3.741657387\times1.54\approx3.741657387\times1.5 = 5.61248608, 3.741657387\times0.04 = 0.149666295, sum is 5.61248608 + 0.149666295 = 5.762152375 \), which is approximately \( 5.8 \) when rounded to the nearest tenth? Wait, no, 5.762 is closer to 5.8? Wait, 5.762, the tenths place is 7, hundredths is 6, so we round up the tenths place: 5.8. Wait, but let's check again. Wait, maybe I mixed up opposite and adjacent. Wait, angle at Q is 57 degrees, right angle at S. So side QS is adjacent to angle Q, side RS is opposite to angle Q, side QR is hypotenuse. So \( \tan(57^\circ)=\frac{RS}{QS} \), so \( RS = QS\times\tan(57^\circ) \). \( QS=\sqrt{14}\approx3.7417 \), \( \tan(57^\circ)\approx1.5399 \), so \( 3.7417\times1.5399\approx3.7417\times1.54\approx5.76 \), which rounds to 5.8? Wait, no, 5.76 is 5.8 when rounded to the nearest tenth? Wait, 5.76: the first decimal is 7, second is 6, so we round up 7 to 8, so 5.8. Wait, but let's calculate more precisely. \( \sqrt{14}=3.7416573867739413 \), \( \tan(57^\circ)=1.5398649638065833 \). Multiply them: \( 3.7416573867739413\times1.5398649638065833 \). Let's do this multiplication:

3.7416573867739413 * 1.5398649638065833:

First, 3 * 1.5398649638065833 = 4.61959489141975

0.7416573867739413 * 1.5398649638065833 ≈ 0.7416573867739413*1.5 = 1.112486080160912, 0.7416573867739413*0.0398649638065833≈0.7416573867739413*0.04=0.02966629547095765, minus 0.7416573867739413*0.0001350361934167≈0.0001002, so approximately 0.029566. So total for 0.741657 part is 1.112486 + 0.029566≈1.142052. Then total RS≈4.61959489 + 1.142052≈5.761647, which is approximately 5.8 when rounded to the nearest tenth? Wait, 5.7616, the tenths digit is 7, hundredths is 6, so we round up the tenths digit: 5.8. Wait, but let's check with calculator steps. Alternatively, maybe I made a mistake in opposite and adjacent. Wait, angle at Q: sides: QS is one leg (adjacent to angle Q), RS is the other leg (opposite to angle Q). So yes, tangent is opposite over adjacent. So formula is correct. So \( RS = \sqrt{14} \times \tan(57^\circ) \). Let's compute \( \sqrt{14}\approx3.7417 \), \( \tan(57^\circ)\approx1.5399 \), so 3.7417*1.5399≈5.76, which is 5.8 when rounded to the nearest tenth. Wait, but maybe I should use more precise values. Let's use calculator:

\( \sqrt{14} \approx 3.74165738677 \)

\( \tan(57^\circ) \approx 1.53986496381 \)

Multiply them: 3.74165738677 * 1.53986496381 = let's do this multiplication:

3.74165738677 * 1.53986496381

= 3.74165738677 * (1 + 0.5 + 0.03 + 0.009 + 0.0008 + 0.00006 + 0.00000496381)

But easier to use calculator: 3.74165738677 * 1.53986496381 ≈ 5.762, which is 5.8 when rounded to the nearest tenth. Wait, but 5.762, the tenths place is 7, hundredths is 6, so we round up the 7 to 8, so 5.8.

Wait, but maybe I messed up the angle. Wait, angle at Q is 57 degrees, right angle at S. So triangle QSR, right-angled at S. So angle Q: 57°, side QS: √14 (adjacent to angle Q), side RS: opposite to angle Q, side QR: hypotenuse. So yes, tangent is opposite over adjacent. So formula is correct. So RS = QS * tan(57°) = √14 * tan(57°) ≈ 3.7417 * 1.5399 ≈ 5.76, which is 5.8 when rounded to the nearest tenth.

Answer:

\( 5.8 \)