Question 1:
Find the length of \(\overline{BC}\)
Solution:
\(\triangle\)ABD is a right angle triangle by itself. This means we can look for \(\overline{AB}\) using tan. Tan takes the opposite over the hypotenuse (Toa in SohCahToa).
\(\tan 22\)° \(= \frac{\overline{AB}}{80}\)
\(\overline{AB}= 32.3 cm\)
Now that we've got \(\overline{AB}\), we can look for \(\overline{AC}\) using tan. \(\triangle\)ACD is also a right angle triangle, so we can use SohCahToa. Again, we're making use of Toa in SohCahToa.
\(\tan 37\)° \(= \frac{\overline{AC}}{80}\)
\(\overline{AC}= 60.3 cm\)
We know the lengths of \(\overline{AB}\) and \(\overline{AC}\), which means we can subtract the length of \(\overline{AB}\) from \(\overline{AC}\) in order to just get the length of \(\overline{BC}\).
\(\overline{BC} = \) \(\overline{AC} - \) \(\overline{AB}\)
\(\overline{BC} = 60.3-32.3\)
= 28 cm
A lot of the questions you'll be given require you to put to use several of the trig ratios. You'll always be working with right triangles, so make sure to keep a note of that when you start the question. For example, this question seemed to require you to use \(\triangle\)BCD at first, but you know that can't be right because it is not a right triangle. Without that 90 degree right angle, you won't be able to use SohCahToa to help you find the unknowns.
Always base your answers around the right triangles in the question and you'll be able to find the answer.
Up for a challenge? Try out this online math test to see if you can answer these trigonometry questions quizzing you on Soh Cah Toa.
A lesson to review that deals with sine, cosine, and tangent includes the ASTC rule. You can also look into these lessons that covers how to find unknowns using the law of sines and the law of cosines. Studying how to find exact values of trigonometric ratios and trig ratios of angles in radians will also advance you further in trig problems