Question

angle 55° 65° 75°
adjacent leg length / hypotenuse length 0.57 0.42 0.26
opposite leg length / hypotenuse length 0.82 0.91 0.97
opposite leg length / adjacent leg length 1.43 2.14 3.73

use the table to approximate ( mangle q ) in the triangle below.

triangle with right angle at ( p ), ( rq = 4.4 ), ( pq = 2.5 ), vertices ( r ), ( p ), ( q )

choose 1 answer:
a ( 55^circ )
b ( 65^circ )
c ( 75^circ )

Explanation:

Step1: Identify the sides relative to ∠Q

In right triangle \( \triangle QPR \) (right - angled at \( P \)), for \( \angle Q \), the opposite side to \( \angle Q \) is \( PR \) with length \( 4.4 \), and the adjacent side to \( \angle Q \) is \( PQ \) with length \( 2.5 \). We can use the tangent ratio, which is \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \).

Step2: Calculate the tangent of ∠Q

Calculate \( \tan(\angle Q)=\frac{PR}{PQ}=\frac{4.4}{2.5} = 1.76 \). Wait, no, wait. Wait, in the triangle, \( PQ = 2.5 \), \( PR=4.4 \)? Wait, no, let's re - check. Wait, the hypotenuse? No, \( \angle P = 90^{\circ} \), so \( PQ \) and \( PR \) are the legs, \( QR \) is the hypotenuse. Wait, for \( \angle Q \), opposite side is \( PR \), adjacent side is \( PQ \). So \( \tan(\angle Q)=\frac{PR}{PQ}=\frac{4.4}{2.5}=1.76 \)? Wait, but the table has \( \frac{\text{opposite leg length}}{\text{adjacent leg length}} \) values. Wait, maybe I mixed up. Wait, the table has three rows: \( \frac{\text{adjacent}}{\text{hypotenuse}} \) (cosine), \( \frac{\text{opposite}}{\text{hypotenuse}} \) (sine), and \( \frac{\text{opposite}}{\text{adjacent}} \) (tangent). Let's recalculate the ratio of opposite to adjacent for \( \angle Q \). Wait, maybe \( PQ = 2.5 \) is adjacent, \( PR = 4.4 \) is opposite? Wait, no, let's check the triangle again. The triangle has right angle at \( P \), so vertices are \( P \) (right angle), \( Q \), and \( R \). So side \( PQ = 2.5 \), side \( PR = 4.4 \), and hypotenuse \( QR \). For angle \( Q \), the adjacent side is \( PQ \) (length \( 2.5 \)), and the opposite side is \( PR \) (length \( 4.4 \)). So \( \frac{\text{opposite}}{\text{adjacent}}=\frac{4.4}{2.5}=1.76 \)? Wait, but the table has values \( 1.43 \) (for \( 55^{\circ} \)), \( 2.14 \) (for \( 65^{\circ} \)), \( 3.73 \) (for \( 75^{\circ} \)). Wait, maybe I got the opposite and adjacent wrong. Wait, maybe \( PQ \) is opposite and \( PR \) is adjacent? Wait, no. Let's think about the angle \( Q \). The angle at \( Q \), so the sides: the side opposite \( Q \) is \( PR \), the side adjacent to \( Q \) is \( PQ \). Wait, but \( 4.4/2.5 = 1.76 \). Let's see the table values: \( 55^{\circ} \) has \( 1.43 \), \( 65^{\circ} \) has \( 2.14 \), \( 75^{\circ} \) has \( 3.73 \). Wait, maybe I made a mistake in identifying the sides. Wait, maybe the hypotenuse is \( 4.4 \)? Wait, the length of \( QR \) is \( 4.4 \), which is the hypotenuse. Then, for angle \( Q \), the adjacent side is \( PQ \), and the hypotenuse is \( QR = 4.4 \), and the opposite side is \( PR \). Let's calculate \( \frac{\text{opposite}}{\text{adjacent}} \) again. Wait, if \( QR = 4.4 \) (hypotenuse), \( PQ = 2.5 \) (adjacent to \( Q \)), then \( PR=\sqrt{QR^{2}-PQ^{2}}=\sqrt{4.4^{2}-2.5^{2}}=\sqrt{19.36 - 6.25}=\sqrt{13.11}\approx3.62 \). Then \( \frac{PR}{PQ}=\frac{3.62}{2.5}\approx1.45 \). Ah, that's close to \( 1.43 \) (the value for \( 55^{\circ} \) in the \( \frac{\text{opposite}}{\text{adjacent}} \) row). Wait, maybe I misread the triangle. Let's re - examine the triangle: the side labeled \( 4.4 \) is \( QR \) (hypotenuse), \( PQ = 2.5 \) (adjacent to \( Q \)). So \( \cos(\angle Q)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{PQ}{QR}=\frac{2.5}{4.4}\approx0.568 \). Looking at the table, the \( \frac{\text{adjacent leg length}}{\text{hypotenuse length}} \) row: for \( 55^{\circ} \) it's \( 0.57 \), which is very close to \( 0.568 \). So that means \( \angle Q\approx55^{\circ} \)? Wait, no, wait. Wait, \( \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} \). So \( \cos(\angle Q)=\frac{PQ}{QR}=\frac{2.5}{4.4}\approx0.568 \). The table has for \( 55^{\circ} \), \( \frac{\text{adjacent}}{\text{hypotenuse}} = 0.57 \), which is a very close match. So \( \angle Q\approx55^{\circ} \)? Wait, but let's check the tangent again. If \( \cos(\angle Q)=0.57 \) (approx \( 55^{\circ} \)), then \( \sin(\angle Q) = 0.82 \) (from the table for \( 55^{\circ} \)). Then \( \sin(\angle Q)=\frac{PR}{QR}\), so \( PR = QR\times\sin(\angle Q)=4.4\times0.82 = 3.608 \). Then \( \tan(\angle Q)=\frac{PR}{PQ}=\frac{3.608}{2.5}\approx1.443 \), which is close to \( 1.43 \) (the table value for \( 55^{\circ} \) in the \( \frac{\text{opposite}}{\text{adjacent}} \) row). So this confirms that \( \angle Q\approx55^{\circ} \). Wait, but initially I thought maybe I mixed up, but after recalculating using cosine (which is adjacent over hypotenuse) and seeing that \( 2.5/4.4\approx0.568 \) is very close to \( 0.57 \) (the value for \( 55^{\circ} \) in the adjacent/hypotenuse row), so the angle is approximately \( 55^{\circ} \).

Answer:

A. \( 55^{\circ} \)