Trigonometry worksheet for finding missing side lengths in right-angled triangles.

Here are the solutions for the missing lengths in the trigonometry worksheet. All answers are rounded to 3 significant figures as requested.

Section A

1)
* Identify sides: Relative to the $35^\circ$ angle, $x$ is the Opposite side and $12$ is the Hypotenuse.
* Formula: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
* Calculation:
$$ \sin(35^\circ) = \frac{x}{12} $$
$$ x = 12 \times \sin(35^\circ) $$
$$ x \approx 6.8829... $$
* Answer: 6.88 cm

2)
* Identify sides: Relative to the $67^\circ$ angle, $x$ is the Adjacent side and $9$ is the Hypotenuse.
* Formula: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
* Calculation:
$$ \cos(67^\circ) = \frac{x}{9} $$
$$ x = 9 \times \cos(67^\circ) $$
$$ x \approx 3.5166... $$
* Answer: 3.52 cm

3)
* Identify sides: Relative to the $49^\circ$ angle, $x$ is the Opposite side and $16$ is the Adjacent side.
* Formula: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
* Calculation:
$$ \tan(49^\circ) = \frac{x}{16} $$
$$ x = 16 \times \tan(49^\circ) $$
$$ x \approx 18.405... $$
* Answer: 18.4 cm

4)
* Identify sides: Relative to the $62^\circ$ angle, $x$ is the Adjacent side and $17$ is the Hypotenuse.
* Formula: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
* Calculation:
$$ \cos(62^\circ) = \frac{x}{17} $$
$$ x = 17 \times \cos(62^\circ) $$
$$ x \approx 7.981... $$
* Answer: 7.98 cm

5)
* Identify sides: Relative to the $27^\circ$ angle (at the top), $x$ is the Opposite side and $5$ is the Adjacent side.
* Formula: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
* Calculation:
$$ \tan(27^\circ) = \frac{x}{5} $$
$$ x = 5 \times \tan(27^\circ) $$
$$ x \approx 2.547... $$
* Answer: 2.55 cm

6)
* Identify sides: Relative to the $33^\circ$ angle, $x$ is the Adjacent side and $21$ is the Hypotenuse.
* Formula: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
* Calculation:
$$ \cos(33^\circ) = \frac{x}{21} $$
$$ x = 21 \times \cos(33^\circ) $$
$$ x \approx 17.612... $$
* Answer: 17.6 cm

---

Section B

1)
* Identify sides: Relative to the $15^\circ$ angle, $14$ is the Adjacent side and $x$ is the Hypotenuse.
* Formula: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
* Calculation:
$$ \cos(15^\circ) = \frac{14}{x} $$
$$ x = \frac{14}{\cos(15^\circ)} $$
$$ x \approx 14.495... $$
* Answer: 14.5 cm

2)
* Identify sides: Relative to the $71^\circ$ angle, $19$ is the Opposite side and $x$ is the Hypotenuse.
* Formula: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
* Calculation:
$$ \sin(71^\circ) = \frac{19}{x} $$
$$ x = \frac{19}{\sin(71^\circ)} $$
$$ x \approx 20.103... $$
* Answer: 20.1 cm

3)
* Identify sides: Relative to the $42^\circ$ angle, $5$ is the Opposite side and $x$ is the Adjacent side.
* Formula: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
* Calculation:
$$ \tan(42^\circ) = \frac{5}{x} $$
$$ x = \frac{5}{\tan(42^\circ)} $$
$$ x \approx 5.553... $$
* Answer: 5.55 cm

4)
* Identify sides: Relative to the $58^\circ$ angle, $11$ is the Adjacent side and $x$ is the Hypotenuse.
* Formula: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
* Calculation:
$$ \cos(58^\circ) = \frac{11}{x} $$
$$ x = \frac{11}{\cos(58^\circ)} $$
$$ x \approx 20.758... $$
* Answer: 20.8 cm

5)
* Identify sides: Relative to the $20^\circ$ angle (at the top), $7$ is the Opposite side and $x$ is the Adjacent side.
* Formula: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
* Calculation:
$$ \tan(20^\circ) = \frac{7}{x} $$
$$ x = \frac{7}{\tan(20^\circ)} $$
$$ x \approx 19.232... $$
* Answer: 19.2 cm

6)
* Identify sides: Relative to the $36^\circ$ angle, $24$ is the Opposite side and $x$ is the Hypotenuse.
* Formula: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
* Calculation:
$$ \sin(36^\circ) = \frac{24}{x} $$
$$ x = \frac{24}{\sin(36^\circ)} $$
$$ x \approx 40.833... $$
* Answer: 40.8 cm

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

Final Answer:
Section A
1) 6.88 cm
2) 3.52 cm
3) 18.4 cm
4) 7.98 cm
5) 2.55 cm
6) 17.6 cm

Section B
1) 14.5 cm
2) 20.1 cm
3) 5.55 cm
4) 20.8 cm
5) 19.2 cm
6) 40.8 cm