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

To solve the problems in the worksheet, we will use trigonometric functions (sine, cosine, and tangent) to find the missing angles or lengths. Let's go through each problem step by step.

Section A: Find the missing angle \( x \)

#### 1)
Given:
- Opposite side = 4 cm
- Hypotenuse = 9 cm

We use the sine function:
\[
\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{9}
\]
\[
x = \arcsin\left(\frac{4}{9}\right)
\]
Using a calculator:
\[
x \approx 26.4^\circ
\]

#### 2)
Given:
- Adjacent side = 10 cm
- Hypotenuse = 14 cm

We use the cosine function:
\[
\cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{10}{14}
\]
\[
x = \arccos\left(\frac{10}{14}\right)
\]
Using a calculator:
\[
x \approx 44.4^\circ
\]

#### 3)
Given:
- Opposite side = 7 cm
- Adjacent side = 12 cm

We use the tangent function:
\[
\tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{7}{12}
\]
\[
x = \arctan\left(\frac{7}{12}\right)
\]
Using a calculator:
\[
x \approx 30.9^\circ
\]

#### 4)
Given:
- Adjacent side = 15 cm
- Hypotenuse = 18 cm

We use the cosine function:
\[
\cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{15}{18}
\]
\[
x = \arccos\left(\frac{15}{18}\right)
\]
Using a calculator:
\[
x \approx 33.6^\circ
\]

#### 5)
Given:
- Opposite side = 0.99 cm
- Adjacent side = 0.83 cm

We use the tangent function:
\[
\tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{0.99}{0.83}
\]
\[
x = \arctan\left(\frac{0.99}{0.83}\right)
\]
Using a calculator:
\[
x \approx 50.2^\circ
\]

#### 6)
Given:
- Opposite side = 410 mm
- Hypotenuse = 972 mm

We use the sine function:
\[
\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{410}{972}
\]
\[
x = \arcsin\left(\frac{410}{972}\right)
\]
Using a calculator:
\[
x \approx 24.1^\circ
\]

Section B: Find the missing angles or lengths \( x \)

#### 1)
Given:
- Adjacent side = 11 cm
- Hypotenuse = 5 cm

We use the cosine function:
\[
\cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{11}{5}
\]
This is not possible because the adjacent side cannot be longer than the hypotenuse. There might be a typo in the problem. Assuming it should be:
- Opposite side = 5 cm
- Adjacent side = 11 cm

We use the tangent function:
\[
\tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{11}
\]
\[
x = \arctan\left(\frac{5}{11}\right)
\]
Using a calculator:
\[
x \approx 24.4^\circ
\]

#### 2)
Given:
- Opposite side = 0.7 cm
- Adjacent side = 0.21 cm

We use the tangent function:
\[
\tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{0.7}{0.21}
\]
\[
x = \arctan\left(\frac{0.7}{0.21}\right)
\]
Using a calculator:
\[
x \approx 71.6^\circ
\]

#### 3)
Given:
- Adjacent side = 6.4 cm
- Angle = 62°

We use the tangent function to find the opposite side:
\[
\tan(62^\circ) = \frac{\text{opposite}}{\text{adjacent}}
\]
\[
\text{opposite} = 6.4 \cdot \tan(62^\circ)
\]
Using a calculator:
\[
\text{opposite} \approx 6.4 \cdot 1.8807 \approx 12.0 \text{ cm}
\]

#### 4)
Given:
- Opposite side = 1.5 cm
- Adjacent side = 27 mm = 2.7 cm

We use the tangent function:
\[
\tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{1.5}{2.7}
\]
\[
x = \arctan\left(\frac{1.5}{2.7}\right)
\]
Using a calculator:
\[
x \approx 29.1^\circ
\]

#### 5)
Given:
- Adjacent side = 2.3 cm
- Angle = 13°

We use the tangent function to find the opposite side:
\[
\tan(13^\circ) = \frac{\text{opposite}}{\text{adjacent}}
\]
\[
\text{opposite} = 2.3 \cdot \tan(13^\circ)
\]
Using a calculator:
\[
\text{opposite} \approx 2.3 \cdot 0.2309 \approx 0.531 \text{ cm}
\]

#### 6)
Given:
- Opposite side = 8 m = 800 cm
- Adjacent side = 620 cm

We use the tangent function:
\[
\tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{800}{620}
\]
\[
x = \arctan\left(\frac{800}{620}\right)
\]
Using a calculator:
\[
x \approx 52.1^\circ
\]

Final Answers:

Section A:
1. \( x \approx 26.4^\circ \)
2. \( x \approx 44.4^\circ \)
3. \( x \approx 30.9^\circ \)
4. \( x \approx 33.6^\circ \)
5. \( x \approx 50.2^\circ \)
6. \( x \approx 24.1^\circ \)

Section B:
1. \( x \approx 24.4^\circ \)
2. \( x \approx 71.6^\circ \)
3. \( x \approx 12.0 \text{ cm} \)
4. \( x \approx 29.1^\circ \)
5. \( x \approx 0.531 \text{ cm} \)
6. \( x \approx 52.1^\circ \)

\[
\boxed{
\begin{array}{ll}
\text{Section A:} & 26.4^\circ, 44.4^\circ, 30.9^\circ, 33.6^\circ, 50.2^\circ, 24.1^\circ \\
\text{Section B:} & 24.4^\circ, 71.6^\circ, 12.0 \text{ cm}, 29.1^\circ, 0.531 \text{ cm}, 52.1^\circ
\end{array}
}
\]