Question

find rs.

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

Explanation:

Step1: Identify the trigonometric ratio

In right triangle \( QSR \), we know the length of the opposite side to angle \( R \) (which is \( QS = 7.1 \)) and we need to find the adjacent side \( RS \). The trigonometric ratio that relates the opposite and adjacent sides of an angle in a right triangle is the tangent function. The formula for tangent is \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). Here, \( \theta = 55^\circ \), opposite side \( = QS = 7.1 \), and adjacent side \( = RS \) (let's call it \( x \)). So we have \( \tan(55^\circ)=\frac{7.1}{x} \).

Step2: Solve for \( x \) (RS)

We can rearrange the formula to solve for \( x \): \( x=\frac{7.1}{\tan(55^\circ)} \). Now we calculate the value of \( \tan(55^\circ) \). Using a calculator, \( \tan(55^\circ)\approx1.4281 \). Then we substitute this value into the formula: \( x = \frac{7.1}{1.4281}\approx4.97\approx5.0 \) (rounded to the nearest tenth). Wait, wait, no, wait. Wait, in the right triangle, angle at \( R \) is \( 55^\circ \), right angle at \( S \), so \( QS \) is opposite to angle \( R \), and \( RS \) is adjacent to angle \( R \). Wait, no, wait: \( QS \) is one leg, \( RS \) is the other leg. Wait, \( QS = 7.1 \), angle at \( R \) is \( 55^\circ \), so \( \tan(55^\circ)=\frac{QS}{RS} \), so \( RS=\frac{QS}{\tan(55^\circ)} \). Let's recalculate: \( \tan(55^\circ)\approx1.4281 \), so \( 7.1\div1.4281\approx4.97 \), which is approximately \( 5.0 \)? Wait, no, wait, maybe I mixed up opposite and adjacent. Wait, angle \( R \): the sides: opposite is \( QS \), adjacent is \( RS \). So \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \), so \( \tan(55^\circ)=\frac{QS}{RS} \), so \( RS = \frac{QS}{\tan(55^\circ)} \). Let's compute \( 7.1\div\tan(55^\circ) \). \( \tan(55) \) is approximately 1.4281. So \( 7.1\div1.4281\approx4.97 \), which is approximately 5.0 when rounded to the nearest tenth? Wait, no, 4.97 is closer to 5.0? Wait, 4.97 rounded to the nearest tenth is 5.0? Wait, no, 4.97: the tenths place is 9, the hundredths place is 7, so we round up the tenths place: 9 + 1 = 10, so we carry over: 4.9 + 0.1 = 5.0. Wait, but let's check with a calculator. Let's do 7.1 divided by tan(55). Let's use a calculator: tan(55) ≈ 1.42814800674. Then 7.1 / 1.42814800674 ≈ 4.971, which is approximately 5.0 when rounded to the nearest tenth? Wait, no, 4.971 rounded to the nearest tenth is 5.0? Wait, 4.971: the tenths digit is 9, the hundredths digit is 7, which is more than 5, so we round up the tenths digit: 9 becomes 10, so we add 1 to the units digit: 4 + 1 = 5, and the tenths digit becomes 0. So 5.0. Wait, but maybe I made a mistake in the ratio. Wait, alternatively, maybe it's cotangent? Wait, no, tangent is opposite over adjacent. Wait, angle at R: the sides: QS is opposite, RS is adjacent. So tan(55) = QS / RS => RS = QS / tan(55). Yes, that's correct. So 7.1 divided by tan(55) ≈ 7.1 / 1.4281 ≈ 4.97 ≈ 5.0. Wait, but let's check with another approach. Alternatively, maybe using cotangent: cot(55) = adjacent / opposite = RS / QS => RS = QS cot(55). Cot(55) is 1 / tan(55) ≈ 0.7002. So 7.1 0.7002 ≈ 4.97 ≈ 5.0. So that's correct.

Answer:

\( 5.0 \)