Right Triangle Trigonometry Lesson - Free Printable

Let's solve each of the right triangles in the image for either x or θ, using trigonometric ratios (sine, cosine, tangent) and the Pythagorean theorem where appropriate. The triangles are not to scale, so we rely on given side lengths and angles.

---

Triangle 1:

- Right triangle with legs: 6 and unknown opposite θ
- Hypotenuse = 11
- We want to find angle θ

We can use cosine since we have adjacent and hypotenuse:

$$
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{6}{11}
$$

$$
\theta = \cos^{-1}\left(\frac{6}{11}\right)
$$

$$
\theta \approx \cos^{-1}(0.5455) \approx 56.9^\circ
$$

✔ Answer: θ ≈ 56.9°

---

Triangle 2:

- Right triangle, one angle = 32°, one leg = 9, other leg = x
- We need to find x, which is adjacent to the 32° angle
- Since it's a right triangle, and we have an angle and one leg, we can use tangent or cosine.

But let’s identify sides:
- Angle = 32°
- Side opposite = 9
- Side adjacent = x
- So:
$$
\tan(32^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{9}{x}
$$

$$
x = \frac{9}{\tan(32^\circ)} \approx \frac{9}{0.6249} \approx 14.4
$$

✔ Answer: x ≈ 14.4

---

Triangle 3:

- Right triangle, hypotenuse = 13, one leg = 9, want to find θ
- Let’s assume θ is the angle opposite the side of length 9.

Then:
$$
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{9}{13}
$$

$$
\theta = \sin^{-1}\left(\frac{9}{13}\right) \approx \sin^{-1}(0.6923) \approx 43.6^\circ
$$

✔ Answer: θ ≈ 43.6°

---

Triangle 4:

- Right triangle, angle = 29°, hypotenuse = 7, want to find x (side opposite 29°)

Use sine:

$$
\sin(29^\circ) = \frac{x}{7}
$$

$$
x = 7 \cdot \sin(29^\circ) \approx 7 \cdot 0.4848 \approx 3.39
$$

✔ Answer: x ≈ 3.39

---

Triangle 5:

- Right triangle, legs = 3 and 6, want to find θ
- Assume θ is the angle opposite the side of length 3.

Then:
$$
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{6} = 0.5
$$

$$
\theta = \tan^{-1}(0.5) \approx 26.6^\circ
$$

✔ Answer: θ ≈ 26.6°

---

Triangle 6:

- Right triangle, angle = 48°, adjacent side = 5, want to find x (opposite side)

Use tangent:

$$
\tan(48^\circ) = \frac{x}{5}
$$

$$
x = 5 \cdot \tan(48^\circ) \approx 5 \cdot 1.1106 \approx 5.55
$$

✔ Answer: x ≈ 5.55

---

✔ Final Answers:

1. θ ≈ 56.9°
2. x ≈ 14.4
3. θ ≈ 43.6°
4. x ≈ 3.39
5. θ ≈ 26.6°
6. x ≈ 5.55

Let me know if you'd like these rounded differently or shown with exact expressions!