Trigonometic Ratios (SOH CAH TOA): Missing Side Lengths Worksheet ... - Free Printable

To solve these trigonometry problems, we need to find the missing side lengths in right-angled triangles. We use the three main trigonometric ratios:

* Sine (sin): $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
* Cosine (cos): $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
* Tangent (tan): $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

How to choose the right ratio:
1. Identify the angle given ($\theta$).
2. Label the sides relative to that angle: Opposite, Adjacent, or Hypotenuse (the longest side, opposite the right angle).
3. Pick the ratio that uses the two sides you have (or are looking for). Remember SOH CAH TOA.
4. Rearrange the formula to solve for $x$.
5. Calculate and round to 3 significant figures.

Here are the step-by-step solutions for every question on the sheet.

---

Section A

1)
* Given: Angle = $35^\circ$, Hypotenuse = $12\text{ cm}$, Opposite = $x$.
* Ratio: SOH $\rightarrow \sin(35^\circ) = \frac{x}{12}$
* Solve: $x = 12 \times \sin(35^\circ)$
* Calculation: $12 \times 0.57357... = 6.8829...$
* Answer: $6.88\text{ cm}$

2)
* Given: Angle = $67^\circ$, Hypotenuse = $9\text{ cm}$, Adjacent = $x$.
* Ratio: CAH $\rightarrow \cos(67^\circ) = \frac{x}{9}$
* Solve: $x = 9 \times \cos(67^\circ)$
* Calculation: $9 \times 0.39073... = 3.5165...$
* Answer: $3.52\text{ cm}$

3)
* Given: Angle = $49^\circ$, Adjacent = $16\text{ cm}$, Opposite = $x$.
* Ratio: TOA $\rightarrow \tan(49^\circ) = \frac{x}{16}$
* Solve: $x = 16 \times \tan(49^\circ)$
* Calculation: $16 \times 1.15036... = 18.4058...$
* Answer: $18.4\text{ cm}$

4)
* Given: Angle = $62^\circ$, Hypotenuse = $17\text{ cm}$, Adjacent = $x$.
* Ratio: CAH $\rightarrow \cos(62^\circ) = \frac{x}{17}$
* Solve: $x = 17 \times \cos(62^\circ)$
* Calculation: $17 \times 0.46947... = 7.9810...$
* Answer: $7.98\text{ cm}$

5)
* Given: Angle = $27^\circ$, Adjacent = $5\text{ cm}$, Opposite = $x$.
* Ratio: TOA $\rightarrow \tan(27^\circ) = \frac{x}{5}$
* Solve: $x = 5 \times \tan(27^\circ)$
* Calculation: $5 \times 0.50952... = 2.5476...$
* Answer: $2.55\text{ cm}$

6)
* Given: Angle = $33^\circ$, Hypotenuse = $21\text{ cm}$, Adjacent = $x$.
* Ratio: CAH $\rightarrow \cos(33^\circ) = \frac{x}{21}$
* Solve: $x = 21 \times \cos(33^\circ)$
* Calculation: $21 \times 0.83867... = 17.612...$
* Answer: $17.6\text{ cm}$

---

Section B

1)
* Given: Angle = $15^\circ$, Adjacent = $14\text{ cm}$, Hypotenuse = $x$.
* Ratio: CAH $\rightarrow \cos(15^\circ) = \frac{14}{x}$
* Solve: $x = \frac{14}{\cos(15^\circ)}$
* Calculation: $14 \div 0.96592... = 14.4938...$
* Answer: $14.5\text{ cm}$

2)
* Given: Angle = $71^\circ$, Opposite = $19\text{ cm}$, Hypotenuse = $x$.
* Ratio: SOH $\rightarrow \sin(71^\circ) = \frac{19}{x}$
* Solve: $x = \frac{19}{\sin(71^\circ)}$
* Calculation: $19 \div 0.94551... = 20.0948...$
* Answer: $20.1\text{ cm}$

3)
* Given: Angle = $42^\circ$, Opposite = $5\text{ cm}$, Adjacent = $x$.
* Ratio: TOA $\rightarrow \tan(42^\circ) = \frac{5}{x}$
* Solve: $x = \frac{5}{\tan(42^\circ)}$
* Calculation: $5 \div 0.90040... = 5.5530...$
* Answer: $5.55\text{ cm}$

4)
* Given: Angle = $58^\circ$, Adjacent = $11\text{ cm}$, Hypotenuse = $x$.
* Ratio: CAH $\rightarrow \cos(58^\circ) = \frac{11}{x}$
* Solve: $x = \frac{11}{\cos(58^\circ)}$
* Calculation: $11 \div 0.52991... = 20.7579...$
* Answer: $20.8\text{ cm}$

5)
* Given: Angle = $20^\circ$, Opposite = $7\text{ cm}$, Adjacent = $x$.
* Ratio: TOA $\rightarrow \tan(20^\circ) = \frac{7}{x}$
* Solve: $x = \frac{7}{\tan(20^\circ)}$
* Calculation: $7 \div 0.36397... = 19.2323...$
* Answer: $19.2\text{ cm}$

6)
* Given: Angle = $36^\circ$, Opposite = $24\text{ cm}$, Hypotenuse = $x$.
* Ratio: SOH $\rightarrow \sin(36^\circ) = \frac{24}{x}$
* Solve: $x = \frac{24}{\sin(36^\circ)}$
* Calculation: $24 \div 0.58778... = 40.8312...$
* Answer: $40.8\text{ cm}$

──────────────────────────────────────

Final Answer:

Section A
1) $x = 6.88\text{ cm}$
2) $x = 3.52\text{ cm}$
3) $x = 18.4\text{ cm}$
4) $x = 7.98\text{ cm}$
5) $x = 2.55\text{ cm}$
6) $x = 17.6\text{ cm}$

Section B
1) $x = 14.5\text{ cm}$
2) $x = 20.1\text{ cm}$
3) $x = 5.55\text{ cm}$
4) $x = 20.8\text{ cm}$
5) $x = 19.2\text{ cm}$
6) $x = 40.8\text{ cm}$