Trigonometic Ratios (SOH CAH TOA): Word Problems (With Clues ... - Free Printable

Let's solve each problem step by step.

---

Problem 1:

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

#### Solution:
- In a right triangle, we can use trigonometric ratios to find missing sides.
- Here, we know:
- AB (opposite side to angle ACB) = 7 cm
- Angle ACB = 64°
- We need to find BC (adjacent side to angle ACB).

Using the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
\[
\tan(64^\circ) = \frac{AB}{BC} = \frac{7}{BC}
\]
Rearranging to solve for BC:
\[
BC = \frac{7}{\tan(64^\circ)}
\]

Using a calculator:
\[
\tan(64^\circ) \approx 2.0503
\]
\[
BC = \frac{7}{2.0503} \approx 3.41 \text{ cm}
\]

Thus, the length of BC is:
\[
\boxed{3.41}
\]

---

Problem 2:

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

#### Solution:
- In a right triangle, one angle is always 90°. The other two angles are complementary (sum to 90°).
- We can use the sine or cosine function to find one of the angles, and then use the fact that the angles sum to 90° to find the other.

Let's denote the angles as follows:
- Angle opposite the side of length 5 cm: \(\theta_1\)
- Angle opposite the side of length 12 cm: \(\theta_2\)
- The right angle is 90°.

Using the sine function for \(\theta_1\):
\[
\sin(\theta_1) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{13}
\]
\[
\theta_1 = \arcsin\left(\frac{5}{13}\right)
\]

Using a calculator:
\[
\theta_1 \approx \arcsin(0.3846) \approx 22.62^\circ
\]

Since the angles in a triangle sum to 180° and one angle is 90°:
\[
\theta_2 = 90^\circ - \theta_1 = 90^\circ - 22.62^\circ \approx 67.38^\circ
\]

Thus, the sizes of the other two angles are:
\[
\boxed{22.6, 67.4}
\]

---

Problem 3:

The perimeter of a right 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:
- The perimeter of the triangle is the sum of its sides:
\[
a + b + c = 24 \text{ cm}
\]
where \(a = 10 \text{ cm}\), \(b = 8 \text{ cm}\), and \(c\) is the hypotenuse.

First, find the hypotenuse \(c\):
\[
10 + 8 + c = 24
\]
\[
c = 24 - 18 = 6 \text{ cm}
\]

Now, we have the sides of the triangle: 6 cm, 8 cm, and 10 cm. This is a right triangle because:
\[
6^2 + 8^2 = 36 + 64 = 100 = 10^2
\]

To find the smallest angle, we use the tangent function. The smallest angle is opposite the smallest side (6 cm). Let \(\theta\) be the smallest angle:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{6}{8} = 0.75
\]
\[
\theta = \arctan(0.75)
\]

Using a calculator:
\[
\theta \approx \arctan(0.75) \approx 36.87^\circ
\]

Thus, the size of the smallest angle is:
\[
\boxed{36.9}
\]

---

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:
- The diagonal of the rectangle forms a right triangle with the length and width of the rectangle.
- Let \(\theta\) be the angle between the diagonal and the longest side (length = 14 cm).
- The width (8 cm) is the opposite side to \(\theta\), and the length (14 cm) is the adjacent side to \(\theta\).

Using the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{8}{14} = \frac{4}{7}
\]
\[
\theta = \arctan\left(\frac{4}{7}\right)
\]

Using a calculator:
\[
\theta \approx \arctan(0.5714) \approx 29.74^\circ
\]

Thus, the angle the diagonal makes with the longest side is:
\[
\boxed{29.7}
\]

---

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:
- The ladder forms a right triangle with the ground and the wall.
- The ladder is the hypotenuse (5 m).
- The angle between the ladder and the ground is 82°.
- We need to find the height up the wall, which is the opposite side to the 82° angle.

Using the sine function:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
\[
\sin(82^\circ) = \frac{\text{height}}{5}
\]
\[
\text{height} = 5 \cdot \sin(82^\circ)
\]

Using a calculator:
\[
\sin(82^\circ) \approx 0.9903
\]
\[
\text{height} = 5 \cdot 0.9903 \approx 4.95 \text{ m}
\]

Thus, the ladder reaches:
\[
\boxed{4.95}
\]

---

Final Answers:

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