Right Triangle Worksheets - Math Monks - Free Printable

To solve the problems in the "Right Triangle Worksheet," we will use trigonometric relationships (sine, cosine, tangent) and the Pythagorean theorem. Let's go through each problem step by step.

---

Problem 1

Given:
- Hypotenuse = 5 cm
- One leg = 4 cm
- Angle to find = \( x \)

Solution:
We can use the cosine function:
\[
\cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5}
\]
\[
x = \cos^{-1}\left(\frac{4}{5}\right)
\]
Using a calculator:
\[
x \approx 36.87^\circ
\]

Answer:
\[
x \approx 36.87^\circ
\]

---

Problem 2

Given:
- One angle = 42°
- Adjacent side = 15 yd
- Opposite side to find = \( x \)

Solution:
We can use the tangent function:
\[
\tan(42^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{15}
\]
\[
x = 15 \cdot \tan(42^\circ)
\]
Using a calculator:
\[
\tan(42^\circ) \approx 0.9004
\]
\[
x \approx 15 \cdot 0.9004 \approx 13.51 \text{ yd}
\]

Answer:
\[
x \approx 13.51 \text{ yd}
\]

---

Problem 3

Given:
- One leg = 12
- Other leg = 5.4
- Angle to find = \( x \)

Solution:
We can use the tangent function:
\[
\tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{5.4}{12}
\]
\[
x = \tan^{-1}\left(\frac{5.4}{12}\right)
\]
Using a calculator:
\[
\tan^{-1}\left(\frac{5.4}{12}\right) \approx 24.20^\circ
\]

Answer:
\[
x \approx 24.20^\circ
\]

---

Problem 4

Given:
- Hypotenuse = 7 in
- One angle = 42°
- Opposite side to find = \( x \)

Solution:
We can use the sine function:
\[
\sin(42^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{x}{7}
\]
\[
x = 7 \cdot \sin(42^\circ)
\]
Using a calculator:
\[
\sin(42^\circ) \approx 0.6691
\]
\[
x \approx 7 \cdot 0.6691 \approx 4.68 \text{ in}
\]

Answer:
\[
x \approx 4.68 \text{ in}
\]

---

Problem 5

Given:
- One angle = 57°
- Adjacent side = 9 m
- Opposite side to find = \( y \)
- Hypotenuse to find = \( z \)

Solution:
1. Find \( y \):
Use the tangent function:
\[
\tan(57^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{y}{9}
\]
\[
y = 9 \cdot \tan(57^\circ)
\]
Using a calculator:
\[
\tan(57^\circ) \approx 1.5399
\]
\[
y \approx 9 \cdot 1.5399 \approx 13.86 \text{ m}
\]

2. Find \( z \):
Use the Pythagorean theorem:
\[
z^2 = 9^2 + y^2
\]
\[
z^2 = 81 + (13.86)^2
\]
\[
z^2 = 81 + 192.10
\]
\[
z^2 \approx 273.10
\]
\[
z \approx \sqrt{273.10} \approx 16.53 \text{ m}
\]

Answers:
\[
y \approx 13.86 \text{ m}, \quad z \approx 16.53 \text{ m}
\]

---

Problem 6

Given:
- One angle = 32°
- Hypotenuse = 11 mm
- Opposite side to find = \( x \)
- Adjacent side to find = \( y \)

Solution:
1. Find \( x \):
Use the sine function:
\[
\sin(32^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{x}{11}
\]
\[
x = 11 \cdot \sin(32^\circ)
\]
Using a calculator:
\[
\sin(32^\circ) \approx 0.5299
\]
\[
x \approx 11 \cdot 0.5299 \approx 5.83 \text{ mm}
\]

2. Find \( y \):
Use the cosine function:
\[
\cos(32^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{y}{11}
\]
\[
y = 11 \cdot \cos(32^\circ)
\]
Using a calculator:
\[
\cos(32^\circ) \approx 0.8480
\]
\[
y \approx 11 \cdot 0.8480 \approx 9.33 \text{ mm}
\]

Answers:
\[
x \approx 5.83 \text{ mm}, \quad y \approx 9.33 \text{ mm}
\]

---

Problem 7

Given:
- One angle = 66°
- Hypotenuse = 9.9 in
- Opposite side to find = \( y \)
- Adjacent side to find = \( x \)
- Another angle to find = \( z \)

Solution:
1. Find \( z \):
Since the triangle is a right triangle, the sum of angles is 180°:
\[
z = 180^\circ - 90^\circ - 66^\circ = 24^\circ
\]

2. Find \( y \):
Use the sine function:
\[
\sin(66^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{9.9}
\]
\[
y = 9.9 \cdot \sin(66^\circ)
\]
Using a calculator:
\[
\sin(66^\circ) \approx 0.9135
\]
\[
y \approx 9.9 \cdot 0.9135 \approx 9.04 \text{ in}
\]

3. Find \( x \):
Use the cosine function:
\[
\cos(66^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{9.9}
\]
\[
x = 9.9 \cdot \cos(66^\circ)
\]
Using a calculator:
\[
\cos(66^\circ) \approx 0.4067
\]
\[
x \approx 9.9 \cdot 0.4067 \approx 4.03 \text{ in}
\]

Answers:
\[
z = 24^\circ, \quad y \approx 9.04 \text{ in}, \quad x \approx 4.03 \text{ in}
\]

---

Problem 8

Given:
- One angle = 47°
- Hypotenuse = 7 yd
- Opposite side to find = \( x \)

Solution:
Use the sine function:
\[
\sin(47^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{x}{7}
\]
\[
x = 7 \cdot \sin(47^\circ)
\]
Using a calculator:
\[
\sin(47^\circ) \approx 0.7314
\]
\[
x \approx 7 \cdot 0.7314 \approx 5.12 \text{ yd}
\]

Answer:
\[
x \approx 5.12 \text{ yd}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ x \approx 36.87^\circ \\
2. & \ x \approx 13.51 \text{ yd} \\
3. & \ x \approx 24.20^\circ \\
4. & \ x \approx 4.68 \text{ in} \\
5. & \ y \approx 13.86 \text{ m}, \ z \approx 16.53 \text{ m} \\
6. & \ x \approx 5.83 \text{ mm}, \ y \approx 9.33 \text{ mm} \\
7. & \ z = 24^\circ, \ y \approx 9.04 \text{ in}, \ x \approx 4.03 \text{ in} \\
8. & \ x \approx 5.12 \text{ yd}
\end{aligned}
}
\]