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

Let's solve each problem step-by-step using trigonometry (SOHCAHTOA), which applies to right-angled triangles.

We'll use:
- SOH: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
- CAH: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
- TOA: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

We’ll also use inverse trigonometric functions: $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$.

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🔷 Section A: Find the missing angle $x$

#### 1)
Right triangle with:
- Opposite = 4 cm
- Hypotenuse = 9 cm

Use SOH:

$$
\sin(x) = \frac{4}{9} \Rightarrow x = \sin^{-1}\left(\frac{4}{9}\right) \approx \sin^{-1}(0.4444) \approx 26.4^\circ
$$

✔ Answer: $x = 26.4^\circ$

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#### 2)
Right triangle with:
- Opposite = 14 cm
- Adjacent = 10 cm

Use TOA:

$$
\tan(x) = \frac{14}{10} = 1.4 \Rightarrow x = \tan^{-1}(1.4) \approx 54.5^\circ
$$

✔ Answer: $x = 54.5^\circ$

---

#### 3)
Right triangle with:
- Opposite = 7 cm
- Adjacent = 12 cm

Use TOA:

$$
\tan(x) = \frac{7}{12} \approx 0.5833 \Rightarrow x = \tan^{-1}(0.5833) \approx 30.2^\circ
$$

✔ Answer: $x = 30.2^\circ$

---

#### 4)
Right triangle with:
- Opposite = 15 cm
- Hypotenuse = 18 cm

Use SOH:

$$
\sin(x) = \frac{15}{18} = 0.8333 \Rightarrow x = \sin^{-1}(0.8333) \approx 56.4^\circ
$$

✔ Answer: $x = 56.4^\circ$

---

#### 5)
Right triangle with:
- Opposite = 0.83 cm
- Adjacent = 0.99 cm

Use TOA:

$$
\tan(x) = \frac{0.83}{0.99} \approx 0.8384 \Rightarrow x = \tan^{-1}(0.8384) \approx 40.1^\circ
$$

✔ Answer: $x = 40.1^\circ$

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#### 6)
Right triangle with:
- Opposite = 410 mm
- Hypotenuse = 972 mm

Use SOH:

$$
\sin(x) = \frac{410}{972} \approx 0.4218 \Rightarrow x = \sin^{-1}(0.4218) \approx 25.0^\circ
$$

✔ Answer: $x = 25.0^\circ$

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🔷 Section B: Find the missing angles or lengths $x$

#### 1)
Right triangle with:
- Opposite = 5 cm
- Adjacent = 11 cm

Use TOA:

$$
\tan(x) = \frac{5}{11} \approx 0.4545 \Rightarrow x = \tan^{-1}(0.4545) \approx 24.4^\circ
$$

✔ Answer: $x = 24.4^\circ$

---

#### 2)
Right triangle with:
- Opposite = 0.21 cm
- Hypotenuse = 0.7 cm

Use SOH:

$$
\sin(x) = \frac{0.21}{0.7} = 0.3 \Rightarrow x = \sin^{-1}(0.3) \approx 17.5^\circ
$$

✔ Answer: $x = 17.5^\circ$

---

#### 3)
Right triangle with:
- Adjacent = 6.4 cm
- Angle = $62^\circ$
- Find hypotenuse $x$

Use CAH:

$$
\cos(62^\circ) = \frac{6.4}{x} \Rightarrow x = \frac{6.4}{\cos(62^\circ)} \approx \frac{6.4}{0.4695} \approx 13.65 \text{ cm}
$$

✔ Answer: $x = 13.7$ cm (to 3 sig fig)

---

#### 4)
Right triangle with:
- Opposite = 1.5 cm
- Hypotenuse = 27 mm = 2.7 cm (convert units!)

Use SOH:

$$
\sin(x) = \frac{1.5}{2.7} \approx 0.5556 \Rightarrow x = \sin^{-1}(0.5556) \approx 33.7^\circ
$$

✔ Answer: $x = 33.7^\circ$

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#### 5)
Right triangle with:
- Adjacent = 2.3 cm
- Angle = $13^\circ$
- Find opposite side $x$

Use TOA:

$$
\tan(13^\circ) = \frac{x}{2.3} \Rightarrow x = 2.3 \times \tan(13^\circ) \approx 2.3 \times 0.2309 \approx 0.531 \text{ cm}
$$

✔ Answer: $x = 0.531$ cm

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#### 6)
Right triangle with:
- Opposite = 8 m = 800 cm
- Adjacent = 620 cm
- Find angle $x$

Use TOA:

$$
\tan(x) = \frac{800}{620} \approx 1.2903 \Rightarrow x = \tan^{-1}(1.2903) \approx 52.2^\circ
$$

✔ Answer: $x = 52.2^\circ$

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✔ Final Answers Summary

#### Section A: Missing Angles
1. $x = 26.4^\circ$
2. $x = 54.5^\circ$
3. $x = 30.2^\circ$
4. $x = 56.4^\circ$
5. $x = 40.1^\circ$
6. $x = 25.0^\circ$

#### Section B: Missing Angles or Lengths
1. $x = 24.4^\circ$
2. $x = 17.5^\circ$
3. $x = 13.7$ cm
4. $x = 33.7^\circ$
5. $x = 0.531$ cm
6. $x = 52.2^\circ$

All answers are rounded to 3 significant figures, as requested.

Let me know if you'd like a diagram explanation or calculator steps!