Question

math 2201 september 2025 7)solve for triangle opq definitions: you will need to understand these angle of elevation- the angle between a horizontal line and the line of sight when looking up at an object angle of depression- the angle between the horizontal line and the line of sight when looking down at an object. word problems: use both the sine, cosine, and soh cah toa, to help solve the following: 1) two security cameras in a museum must be adjusted to monitor a new display of fossils. the cameras are mounted 6 m above the floor, directly across from each other on opposite walls. the walls are 12 m apart. the fossils are displayed in cases made of wood and glass. the top of the display is 1.5 m above the floor. the distance from the camera on the left to the center of the top of the display is 4.8 m. determine the angle of depression, to the nearest degree for each camera.

Explanation:

Step1: Recall the Law of Cosines

The Law of Cosines is \(c^{2}=a^{2}+b^{2}-2ab\cos C\). In \(\triangle OPQ\) (assuming \(a = 14.4\), \(b=30.6\), and \(C = 46^{\circ}\)), we first find the length of the third - side. Let the side opposite the \(46^{\circ}\) angle be \(c\).
\[c^{2}=14.4^{2}+30.6^{2}-2\times14.4\times30.6\times\cos46^{\circ}\]
\[c^{2}=207.36 + 936.36-890.88\times0.6947\]
\[c^{2}=207.36 + 936.36 - 618.47\]
\[c^{2}=525.25\]
\[c=\sqrt{525.25}\approx22.92\]

Step2: Use the Law of Sines to find an angle

The Law of Sines is \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\). Let's find angle \(B\).
\(\frac{\sin B}{30.6}=\frac{\sin46^{\circ}}{22.92}\)
\(\sin B=\frac{30.6\times\sin46^{\circ}}{22.92}\)
\(\sin B=\frac{30.6\times0.6947}{22.92}\)
\(\sin B=\frac{21.258}{22.92}\approx0.9275\)
\(B=\sin^{- 1}(0.9275)\approx68^{\circ}\)

Step3: Find the third angle

Since the sum of angles in a triangle is \(180^{\circ}\), let the third angle be \(A\).
\(A = 180^{\circ}-46^{\circ}-68^{\circ}=66^{\circ}\)

Answer:

The side - length opposite the \(46^{\circ}\) angle is approximately \(22.9\), one of the other angles is approximately \(68^{\circ}\), and the third angle is approximately \(66^{\circ}\)