SOH-CAH-TOA, solving right triangles, finding angles with inverse functions, and angles of elevation and depression in real-world problems.
SOH-CAH-TOA
For an acute angle \(\theta\) in a right triangle, name the sides relative to \(\theta\): the opposite side, the adjacent side (the leg that touches \(\theta\)) and the hypotenuse.
These agree with the unit-circle definitions: scale the triangle so the hypotenuse is \(1\), and the legs become the cosine and sine.
Solving right triangles
Solving a triangle means finding all its sides and angles. For a right triangle, two pieces of information are enough, as long as at least one is a side.
Angles of elevation and depression
Because the horizontal lines at the observer and at the object are parallel, the angle of depression from \(A\) to \(B\) equals the angle of elevation from \(B\) to \(A\) (alternate interior angles).
Practice
Quick check
Which ratio is \(\cos\theta\)?
CAH: cosine is adjacent over hypotenuse.
Quick check
The angle of depression from a cliff top to a boat is \(25°\). What is the angle of elevation from the boat to the cliff top?
The two angles are alternate interior angles between parallel horizontal lines.
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