Right Triangle Trigonometry Worksheets - Full Set (Free Download) - Free Printable

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

---

Problem 1:

Given:
- A right triangle with one leg = 38, hypotenuse = 86.
- We need to find the other leg \( x \).

#### Solution:
Using the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
where \( a = 38 \), \( b = x \), and \( c = 86 \).

\[
38^2 + x^2 = 86^2
\]
\[
1444 + x^2 = 7396
\]
\[
x^2 = 7396 - 1444
\]
\[
x^2 = 5952
\]
\[
x = \sqrt{5952}
\]
\[
x = 4\sqrt{372} \quad \text{(or approximately 77.15)}
\]

Answer:
\[
x = 4\sqrt{372}
\]

---

Problem 2:

Given:
- A right triangle with one leg = 12, the other leg = 17.
- We need to find the hypotenuse \( x \).

#### Solution:
Using the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
where \( a = 12 \), \( b = 17 \), and \( c = x \).

\[
12^2 + 17^2 = x^2
\]
\[
144 + 289 = x^2
\]
\[
433 = x^2
\]
\[
x = \sqrt{433}
\]

Answer:
\[
x = \sqrt{433}
\]

---

Problem 3:

Given:
- A right triangle with one leg = \( \frac{b}{2} \), hypotenuse = \( b \).
- We need to find the other leg \( x \).

#### Solution:
Using the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
where \( a = x \), \( b = \frac{b}{2} \), and \( c = b \).

\[
x^2 + \left(\frac{b}{2}\right)^2 = b^2
\]
\[
x^2 + \frac{b^2}{4} = b^2
\]
Multiply through by 4 to clear the fraction:
\[
4x^2 + b^2 = 4b^2
\]
\[
4x^2 = 3b^2
\]
\[
x^2 = \frac{3b^2}{4}
\]
\[
x = \frac{\sqrt{3}b}{2}
\]

Answer:
\[
x = \frac{\sqrt{3}b}{2}
\]

---

Problem 4:

Given:
- A right triangle with one angle = 30°, the side opposite the 30° angle = 5/3.
- We need to find the hypotenuse \( x \).

#### Solution:
In a 30-60-90 triangle, the side opposite the 30° angle is half the hypotenuse. Therefore:
\[
\text{Hypotenuse} = 2 \times \text{side opposite 30°}
\]
\[
x = 2 \times \frac{5}{3}
\]
\[
x = \frac{10}{3}
\]

Answer:
\[
x = \frac{10}{3}
\]

---

Problem 5:

Given:
- A right triangle with one angle = 45°, one leg = 32.
- We need to find the other leg \( x \).

#### Solution:
In a 45-45-90 triangle, both legs are equal. Therefore:
\[
x = 32
\]

Answer:
\[
x = 32
\]

---

Problem 6:

Given:
- A right triangle with one angle = 60°, the side opposite the 60° angle = 15.
- We need to find the hypotenuse \( x \).

#### Solution:
In a 30-60-90 triangle, the side opposite the 60° angle is \( \sqrt{3} \) times the side opposite the 30° angle. The hypotenuse is twice the side opposite the 30° angle.

First, find the side opposite the 30° angle:
\[
\text{Side opposite 30°} = \frac{\text{Side opposite 60°}}{\sqrt{3}} = \frac{15}{\sqrt{3}} = 5\sqrt{3}
\]

Now, find the hypotenuse:
\[
\text{Hypotenuse} = 2 \times \text{Side opposite 30°} = 2 \times 5\sqrt{3} = 10\sqrt{3}
\]

Answer:
\[
x = 10\sqrt{3}
\]

---

Problem 7:

Given:
- A right triangle with one angle = 10°, the adjacent side = 15.
- We need to find the opposite side \( x \).

#### Solution:
Use the tangent function:
\[
\tan(10°) = \frac{\text{opposite}}{\text{adjacent}}
\]
\[
\tan(10°) = \frac{x}{15}
\]
\[
x = 15 \cdot \tan(10°)
\]

Using a calculator:
\[
\tan(10°) \approx 0.1763
\]
\[
x \approx 15 \cdot 0.1763 \approx 2.6445
\]

Answer:
\[
x \approx 2.64
\]

---

Problem 8:

Given:
- A right triangle with one angle = 64°, the adjacent side = 15.
- We need to find the opposite side \( y \).

#### Solution:
Use the tangent function:
\[
\tan(64°) = \frac{\text{opposite}}{\text{adjacent}}
\]
\[
\tan(64°) = \frac{y}{15}
\]
\[
y = 15 \cdot \tan(64°)
\]

Using a calculator:
\[
\tan(64°) \approx 2.0503
\]
\[
y \approx 15 \cdot 2.0503 \approx 30.7545
\]

Answer:
\[
y \approx 30.75
\]

---

Problem 9:

Given:
- A right triangle with one angle = 35°, the hypotenuse = \( \sqrt{4} \).
- We need to find the opposite side \( y \).

#### Solution:
Use the sine function:
\[
\sin(35°) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
\[
\sin(35°) = \frac{y}{\sqrt{4}}
\]
\[
\sin(35°) = \frac{y}{2}
\]
\[
y = 2 \cdot \sin(35°)
\]

Using a calculator:
\[
\sin(35°) \approx 0.5736
\]
\[
y \approx 2 \cdot 0.5736 \approx 1.1472
\]

Answer:
\[
y \approx 1.15
\]

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & x = 4\sqrt{372} \\
2. & x = \sqrt{433} \\
3. & x = \frac{\sqrt{3}b}{2} \\
4. & x = \frac{10}{3} \\
5. & x = 32 \\
6. & x = 10\sqrt{3} \\
7. & x \approx 2.64 \\
8. & y \approx 30.75 \\
9. & y \approx 1.15 \\
\end{array}
}
\]