Trigonometry word problems worksheet for GCSE Foundation/Higher tier, featuring five questions on calculating lengths, angles, and applications of trigonometric principles.

Let's solve each problem step by step.

---

Problem 1:

ABC is a right-angled triangle. AB = 7 cm, angle ABC = 90°, and angle ACB = 64°. Calculate the length of BC.

#### Solution:
1. Identify the sides:
- AB is the side opposite to angle ACB.
- BC is the adjacent side to angle ACB.
- AC is the hypotenuse.

2. Use the tangent function:
\[
\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}
\]
Here, \(\tan(64^\circ) = \frac{AB}{BC}\).

3. Substitute the known values:
\[
\tan(64^\circ) = \frac{7}{BC}
\]

4. Solve for \(BC\):
\[
BC = \frac{7}{\tan(64^\circ)}
\]

5. Calculate \(\tan(64^\circ)\):
\[
\tan(64^\circ) \approx 2.0503
\]

6. Substitute and solve:
\[
BC = \frac{7}{2.0503} \approx 3.41
\]

#### Final Answer:
\[
\boxed{3.41}
\]

---

Problem 2:

The lengths of the sides of a right-angled triangle are 5 cm, 12 cm, and 13 cm. Work out the size of the other two angles of this triangle.

#### Solution:
1. Identify the sides:
- The hypotenuse is the longest side, which is 13 cm.
- The other two sides are 5 cm and 12 cm.

2. Use the sine or cosine function to find one of the angles:
- Let \(\theta\) be the angle opposite the side of length 5 cm.
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{13}
\]

3. Solve for \(\theta\):
\[
\theta = \sin^{-1}\left(\frac{5}{13}\right)
\]

4. Calculate \(\sin^{-1}\left(\frac{5}{13}\right)\):
\[
\theta \approx 22.62^\circ
\]

5. Find the other non-right angle:
- The sum of angles in a triangle is \(180^\circ\).
- The right angle is \(90^\circ\).
- Let the other angle be \(\phi\):
\[
\phi = 180^\circ - 90^\circ - 22.62^\circ = 67.38^\circ
\]

#### Final Answers:
\[
\boxed{22.6, 67.4}
\]

---

Problem 3:

The perimeter of a right-angled triangle is 24 cm. The length of one of the sides is 10 cm, the length of the other side is 8 cm. Calculate the size of the smallest angle of this triangle.

#### Solution:
1. Identify the sides:
- Perimeter = 10 + 8 + hypotenuse = 24 cm.
- Hypotenuse = \(24 - 10 - 8 = 6\) cm.

2. Verify if it forms a right-angled triangle using the Pythagorean theorem:
\[
10^2 + 8^2 = 100 + 64 = 164 \quad \text{(not equal to } 6^2 = 36\text{)}
\]
This indicates an error in the problem setup. However, assuming the given sides are correct, we proceed with the given information.

3. Use the tangent function to find the smallest angle:
- The smallest angle is opposite the shortest side (6 cm).
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{6}{8} = 0.75
\]

4. Solve for \(\theta\):
\[
\theta = \tan^{-1}(0.75)
\]

5. Calculate \(\tan^{-1}(0.75)\):
\[
\theta \approx 36.87^\circ
\]

#### Final Answer:
\[
\boxed{36.9}
\]

---

Problem 4:

A rectangle has a length of 14 cm and width of 8 cm. What angle does the diagonal make with the longest side?

#### Solution:
1. Identify the sides:
- Length = 14 cm.
- Width = 8 cm.
- Diagonal can be found using the Pythagorean theorem:
\[
\text{Diagonal} = \sqrt{14^2 + 8^2} = \sqrt{196 + 64} = \sqrt{260} \approx 16.12 \text{ cm}
\]

2. Use the tangent function to find the angle:
- Let \(\theta\) be the angle between the diagonal and the longest side (length).
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{8}{14} = \frac{4}{7}
\]

3. Solve for \(\theta\):
\[
\theta = \tan^{-1}\left(\frac{4}{7}\right)
\]

4. Calculate \(\tan^{-1}\left(\frac{4}{7}\right)\):
\[
\theta \approx 29.74^\circ
\]

#### Final Answer:
\[
\boxed{29.7}
\]

---

Problem 5:

A ladder is 5 m long. The ladder rests against a vertical wall. The foot of the ladder rests on horizontal ground. The ladder makes an angle of 82° with the ground. How far up the wall does the ladder reach?

#### Solution:
1. Identify the sides:
- The ladder is the hypotenuse (5 m).
- The height up the wall is the opposite side.
- The angle between the ladder and the ground is 82°.

2. Use the sine function:
\[
\sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
\[
\sin(82^\circ) = \frac{\text{height}}{5}
\]

3. Solve for the height:
\[
\text{height} = 5 \cdot \sin(82^\circ)
\]

4. Calculate \(\sin(82^\circ)\):
\[
\sin(82^\circ) \approx 0.9903
\]

5. Substitute and solve:
\[
\text{height} = 5 \cdot 0.9903 \approx 4.95
\]

#### Final Answer:
\[
\boxed{4.95}
\]

---

Final Answers:

1. \(\boxed{3.41}\)
2. \(\boxed{22.6, 67.4}\)
3. \(\boxed{36.9}\)
4. \(\boxed{29.7}\)
5. \(\boxed{4.95}\)