Question

find qr.
write your answer as an integer or as a decimal rounded to
qr =
submit

Explanation:

Step1: Identify the triangle type and trigonometric ratio

We have a right - triangle \( \triangle PQR \) with \( \angle P = 90^{\circ} \), \( \angle Q=66^{\circ} \), and \( PQ = 0.9 \). We want to find the length of the hypotenuse \( QR \). In a right - triangle, the cosine of an angle is defined as the adjacent side divided by the hypotenuse. For \( \angle Q \), the adjacent side is \( PQ \) and the hypotenuse is \( QR \). So, \( \cos(Q)=\frac{PQ}{QR} \).

Step2: Solve for \( QR \)

We know that \( \cos(66^{\circ})\approx0.4067 \) and \( PQ = 0.9 \). From the formula \( \cos(Q)=\frac{PQ}{QR} \), we can re - arrange it to get \( QR=\frac{PQ}{\cos(Q)} \). Substituting the values, we have \( QR=\frac{0.9}{\cos(66^{\circ})} \). Plugging in \( \cos(66^{\circ})\approx0.4067 \), we get \( QR=\frac{0.9}{0.4067}\approx2.21 \) (rounded to two decimal places) or if we round to a reasonable decimal place, we can also calculate it more accurately. Let's use a calculator for \( \cos(66^{\circ}) \): \( \cos(66^{\circ})\approx0.4067366431 \). Then \( QR=\frac{0.9}{0.4067366431}\approx2.21 \). If we round to one decimal place, it is approximately \( 2.2 \), but let's check the calculation again. Wait, maybe we made a mistake in the trigonometric ratio. Wait, in right - triangle \( PQR \), \( \angle Q = 66^{\circ} \), \( PQ \) is adjacent to \( \angle Q \), and \( QR \) is the hypotenuse. So \( \cos(66^{\circ})=\frac{PQ}{QR}\Rightarrow QR = \frac{PQ}{\cos(66^{\circ})}\). Let's recalculate: \( \frac{0.9}{0.4067366431}\approx2.21 \). If we use more precise calculation: \( 0.9\div\cos(66^{\circ})=0.9\div0.4067366431\approx2.21 \).

Answer:

\( 2.21 \) (or if we round to a different decimal place, but based on the calculation, approximately \( 2.2 \) or \( 2.21 \))