Trigonometry worksheet for finding missing lengths in right-angled triangles, featuring 12 problems with angles and side measurements.

Here are the solutions for the missing lengths in the triangles. I have calculated each one using trigonometry (SOH CAH TOA) and rounded the answers to 3 significant figures as requested.

Section A

1)
* Identify sides: We have the hypotenuse (12 cm) and want the side opposite the angle ($x$). The angle is $35^\circ$.
* Ratio: Sine ($\sin = \frac{\text{Opposite}}{\text{Hypotenuse}}$).
* Calculation:
$$x = 12 \times \sin(35^\circ)$$
$$x \approx 6.8829$$
* Answer: $x = 6.88$ cm

2)
* Identify sides: We have the hypotenuse (9 cm) and want the side adjacent to the angle ($x$). The angle is $67^\circ$.
* Ratio: Cosine ($\cos = \frac{\text{Adjacent}}{\text{Hypotenuse}}$).
* Calculation:
$$x = 9 \times \cos(67^\circ)$$
$$x \approx 3.5166$$
* Answer: $x = 3.52$ cm

3)
* Identify sides: We have the side adjacent to the angle (16 cm) and want the side opposite ($x$). The angle is $49^\circ$.
* Ratio: Tangent ($\tan = \frac{\text{Opposite}}{\text{Adjacent}}$).
* Calculation:
$$x = 16 \times \tan(49^\circ)$$
$$x \approx 18.405$$
* Answer: $x = 18.4$ cm

4)
* Identify sides: We have the hypotenuse (17 cm) and want the side adjacent to the angle ($x$). The angle is $62^\circ$.
* Ratio: Cosine ($\cos = \frac{\text{Adjacent}}{\text{Hypotenuse}}$).
* Calculation:
$$x = 17 \times \cos(62^\circ)$$
$$x \approx 7.981$$
* Answer: $x = 7.98$ cm

5)
* Identify sides: We have the side opposite the angle (5 cm) and want the hypotenuse ($x$). The angle is $27^\circ$.
* Ratio: Sine ($\sin = \frac{\text{Opposite}}{\text{Hypotenuse}}$).
* Calculation:
$$\sin(27^\circ) = \frac{5}{x} \rightarrow x = \frac{5}{\sin(27^\circ)}$$
$$x \approx 11.013$$
* Answer: $x = 11.0$ cm

6)
* Identify sides: We have the hypotenuse (21 cm) and want the side opposite the angle ($x$). The angle is $33^\circ$.
* Ratio: Sine ($\sin = \frac{\text{Opposite}}{\text{Hypotenuse}}$).
* Calculation:
$$x = 21 \times \sin(33^\circ)$$
$$x \approx 11.438$$
* Answer: $x = 11.4$ cm

---

Section B

1)
* Identify sides: We have the side adjacent to the angle (14 cm) and want the hypotenuse ($x$). The angle is $19^\circ$.
* Ratio: Cosine ($\cos = \frac{\text{Adjacent}}{\text{Hypotenuse}}$).
* Calculation:
$$\cos(19^\circ) = \frac{14}{x} \rightarrow x = \frac{14}{\cos(19^\circ)}$$
$$x \approx 14.825$$
* Answer: $x = 14.8$ cm

2)
* Identify sides: We have the hypotenuse (19 cm) and want the side opposite the angle ($x$). The angle is $77^\circ$.
* Ratio: Sine ($\sin = \frac{\text{Opposite}}{\text{Hypotenuse}}$).
* Calculation:
$$x = 19 \times \sin(77^\circ)$$
$$x \approx 18.511$$
* Answer: $x = 18.5$ cm

3)
* Identify sides: We have the side opposite the angle (5 cm) and want the side adjacent ($x$). The angle is $47^\circ$.
* Ratio: Tangent ($\tan = \frac{\text{Opposite}}{\text{Adjacent}}$).
* Calculation:
$$\tan(47^\circ) = \frac{5}{x} \rightarrow x = \frac{5}{\tan(47^\circ)}$$
$$x \approx 4.662$$
* Answer: $x = 4.66$ cm

4)
* Identify sides: We have the hypotenuse (11 cm) and want the side adjacent to the angle ($x$). The angle is $59^\circ$.
* Ratio: Cosine ($\cos = \frac{\text{Adjacent}}{\text{Hypotenuse}}$).
* Calculation:
$$x = 11 \times \cos(59^\circ)$$
$$x \approx 5.665$$
* Answer: $x = 5.67$ cm

5)
* Identify sides: We have the side adjacent to the angle (7 cm) and want the side opposite ($x$). The angle is $70^\circ$.
* Ratio: Tangent ($\tan = \frac{\text{Opposite}}{\text{Adjacent}}$).
* Calculation:
$$x = 7 \times \tan(70^\circ)$$
$$x \approx 19.230$$
* Answer: $x = 19.2$ cm

6)
* Identify sides: We have the hypotenuse (24 cm) and want the side adjacent to the angle ($x$). The angle is $36^\circ$.
* Ratio: Cosine ($\cos = \frac{\text{Adjacent}}{\text{Hypotenuse}}$).
* Calculation:
$$x = 24 \times \cos(36^\circ)$$
$$x \approx 19.416$$
* Answer: $x = 19.4$ cm

Final Answer:
Section A
1) 6.88 cm
2) 3.52 cm
3) 18.4 cm
4) 7.98 cm
5) 11.0 cm
6) 11.4 cm

Section B
1) 14.8 cm
2) 18.5 cm
3) 4.66 cm
4) 5.67 cm
5) 19.2 cm
6) 19.4 cm