Question

using cos:
28
32°
x
42
57°
x
12
8
x°
using tan:
x
61°
43
x
57°
48
17
x°
52

Explanation:

Step1: Solve first "Using Cos" triangle (left)

In a right - triangle, the cosine of an angle is defined as the adjacent side divided by the hypotenuse. For the left - most triangle under "Using Cos", we have an angle of \(32^{\circ}\), the hypotenuse \(c = 28\), and the adjacent side to the \(32^{\circ}\) angle is \(x\). So, \(\cos(32^{\circ})=\frac{x}{28}\). Then \(x = 28\times\cos(32^{\circ})\). Calculating \(\cos(32^{\circ})\approx0.8480\), so \(x\approx28\times0.8480 = 23.744\).

Step2: Solve second "Using Cos" triangle (middle)

For the middle triangle under "Using Cos", we have an angle of \(57^{\circ}\), the adjacent side to the \(57^{\circ}\) angle is \(42\), and the hypotenuse is \(x\). Using the cosine formula \(\cos(57^{\circ})=\frac{42}{x}\), we can re - arrange it to \(x=\frac{42}{\cos(57^{\circ})}\). \(\cos(57^{\circ})\approx0.5446\), so \(x\approx\frac{42}{0.5446}\approx77.12\).

Step3: Solve third "Using Cos" triangle (right)

For the right - most triangle under "Using Cos", we have the hypotenuse \(c = 12\) and the adjacent side to the angle \(x^{\circ}\) is \(8\). Using the cosine formula \(\cos(x^{\circ})=\frac{8}{12}=\frac{2}{3}\). Then \(x^{\circ}=\cos^{- 1}(\frac{2}{3})\approx48.19^{\circ}\).

Step4: Solve first "Using Tan" triangle (left)

In a right - triangle, the tangent of an angle is defined as the opposite side divided by the adjacent side. For the left - most triangle under "Using Tan", we have an angle of \(61^{\circ}\), the adjacent side \(a = 43\), and the opposite side is \(x\). So, \(\tan(61^{\circ})=\frac{x}{43}\). Then \(x = 43\times\tan(61^{\circ})\). Since \(\tan(61^{\circ})\approx1.8040\), \(x\approx43\times1.8040 = 77.572\).

Step5: Solve second "Using Tan" triangle (middle)

For the middle triangle under "Using Tan", we have an angle of \(57^{\circ}\), the opposite side to the \(57^{\circ}\) angle is \(x\), and the adjacent side is \(48\). Using the tangent formula \(\tan(57^{\circ})=\frac{x}{48}\). Since \(\tan(57^{\circ})\approx1.5399\), then \(x = 48\times1.5399\approx73.915\).

Step6: Solve third "Using Tan" triangle (right)

For the right - most triangle under "Using Tan", we have the opposite side \(o = 52\) and the adjacent side \(a = 17\). Using the tangent formula \(\tan(x^{\circ})=\frac{52}{17}\approx3.0588\). Then \(x^{\circ}=\tan^{-1}(3.0588)\approx71.89^{\circ}\).

Answer:

• "Using Cos" (left): \(x\approx23.74\)
• "Using Cos" (middle): \(x\approx77.12\)
• "Using Cos" (right): \(x\approx48.19^{\circ}\)
• "Using Tan" (left): \(x\approx77.57\)
• "Using Tan" (middle): \(x\approx73.92\)
• "Using Tan" (right): \(x\approx71.89^{\circ}\)