SOH CAH TOA, and when to divide

Lesson 2 · right-triangle word problems · about 8 minutes · fixes Version 3, questions 5 and 11

Every one of these problems comes down to two decisions: which ratio (set by the two sides involved) and multiply or divide (set by where the unknown lands).

The ratios, from Lippman & Rasmussen §5.5:

\[\textbf{S}\text{in}=\frac{\textbf{O}\text{pposite}}{\textbf{H}\text{ypotenuse}}\qquad \textbf{C}\text{os}=\frac{\textbf{A}\text{djacent}}{\textbf{H}\text{ypotenuse}}\qquad \textbf{T}\text{an}=\frac{\textbf{O}\text{pposite}}{\textbf{A}\text{djacent}}\]

The method

  1. Find the angle you were given (never the \(90°\) one).
  2. Label the sides from that angle: H is across from the right angle (the longest, slanted side). O is across from your angle (the missing-letter rule again). A touches your angle and isn't H.
  3. Pick the ratio from the two sides in play, the one you know and the one you want: O+H → sin, A+H → cos, O+A → tan.
  4. Write it and solve. Unknown on top → multiply. Unknown on bottom → divide.
  5. Sanity check: the hypotenuse must be the longest side.

Question 5, worked (the ramp)

Version 3 · Q5

A ramp rises to a doorway \(3\) ft high and makes a \(5°\) angle with the ground. How long is the ramp?

5° O = 3 H = ramp = L ? A
Know O, want H → sine.

\[\sin5°=\frac{3}{L}\]

The unknown \(L\) is on the bottom. Multiply both sides by \(L\), then divide by \(\sin5°\):

\[L\sin5°=3\quad\Rightarrow\quad L=\frac{3}{\sin5°}\]

Sanity check: \(\sin\) of any angle is less than 1, so \(3\sin5°\) is less than 3. A ramp shorter than its own height is impossible, which rules out that answer immediately. (For comparison, the textbook's Example 3 does the same move: \(\sin30°=\frac{7}{b}\Rightarrow b=\frac{7}{\sin30°}=14\).)

Question 11, worked (the ladder)

Version 3 · Q11

A \(20\)-ft ladder makes a \(70°\) angle with the ground. How far is its base from the wall?

70° H = 20 A = d ? O (wall)
Know H, want A → cosine.

\[\cos70°=\frac{d}{20}\]

The unknown \(d\) is on top, so multiply: \(d=20\cos70°\).

The tempting wrong answer is \(20\sin70°\). That's the height up the wall, the side across from the \(70°\) angle, not the side touching it.

What to watch for with your student. Make him label H, O, A on a sketch before writing sin, cos or tan. Most wrong answers come from choosing a ratio by feel. Then ask one question: "is the unknown on the top or the bottom?"

Your turn

Sketch and label each one on paper first. The last question is a review of Lesson 1.