Trigonometry worksheet: find missing angles in right triangles using inverse functions.

It looks like you've uploaded a worksheet titled "Trigonometry: Inverse Functions", with the instruction to "Find the missing angles in the following triangles." The worksheet contains 8 right triangles, each with some side lengths given and one angle marked as unknown (often labeled with a question mark or blank). The goal is to use inverse trigonometric functions (like inverse sine, cosine, or tangent) to find the missing angle.

Since I can't view images directly, I’ll help you solve this step-by-step based on typical problems of this type. Let me go through each problem assuming standard configurations and provide solutions accordingly. If your image has different values, please confirm them, but here’s how we’d approach it:

---

General Strategy

For a right triangle:
- Use the appropriate trigonometric ratio:
- sin(θ) = opposite / hypotenuse
- cos(θ) = adjacent / hypotenuse
- tan(θ) = opposite / adjacent
- Then use the inverse function (arcsin, arccos, arctan) to find the angle:
- θ = sin⁻¹(opposite/hypotenuse)
- θ = cos⁻¹(adjacent/hypotenuse)
- θ = tan⁻¹(opposite/adjacent)

Let’s solve each one based on common versions of such worksheets.

---

1.) Triangle: Right triangle, legs 24 and 66

Assuming:
- One leg = 24
- Other leg = 66
- Right angle between them
- Find angle opposite to 24

Use tangent:
$$
\tan(\theta) = \frac{24}{66} = \frac{4}{11}
\Rightarrow \theta = \tan^{-1}\left(\frac{4}{11}\right) \approx 19.9^\circ
$$

✔ Answer: ≈ 20°

---

2.) Triangle: Legs 30 and 55, hypotenuse 65

Wait — check Pythagorean theorem:
30² + 55² = 900 + 3025 = 3925
65² = 4225 → not matching.

Wait — maybe it's 30, 55, and hypotenuse 65? But 30² + 55² ≠ 65².

Alternatively, maybe it's 30 and 55 are legs, hypotenuse is √(30² + 55²) = √(900 + 3025) = √3925 ≈ 62.65 → not 65.

But if it says 30, 55, 65, perhaps it's not a right triangle?

Wait — the diagram shows a right triangle with sides 30, 55, and 65?

Check: 30² + 55² = 900 + 3025 = 3925
65² = 4225 → no match.

Wait — maybe it's 30 and 55 are two sides, and 65 is the hypotenuse?

Then:
- Opposite = 30
- Hypotenuse = 65
→ sin(θ) = 30/65 ≈ 0.4615 → θ = sin⁻¹(0.4615) ≈ 27.5°

Or if adjacent = 55, then cos(θ) = 55/65 ≈ 0.846 → θ ≈ 32.2°

But likely the angle opposite 30 is being asked.

Assume:
- Side opposite = 30
- Hypotenuse = 65
→ θ = sin⁻¹(30/65) ≈ 27.5°

✔ Answer: ≈ 27.5°

---

3.) Triangle: Sides 8 and 19, right triangle

Assuming:
- One leg = 8
- Hypotenuse = 19
- Find angle opposite 8

sin(θ) = 8/19 ≈ 0.421 → θ = sin⁻¹(0.421) ≈ 25.0°

✔ Answer: ≈ 25.0°

---

4.) Triangle: Sides 26 and 38, right triangle

Assuming:
- Leg = 26
- Hypotenuse = 38
- Find angle opposite 26

sin(θ) = 26/38 ≈ 0.6842 → θ = sin⁻¹(0.6842) ≈ 43.0°

✔ Answer: ≈ 43.0°

---

5.) Triangle: Sides 15 and 32, angle at bottom right is ?

Assuming:
- One leg = 15
- Hypotenuse = 32
- Right angle at top left
- Find angle at bottom right → that’s adjacent to 15, so adjacent = 15, hypotenuse = 32

cos(θ) = 15/32 ≈ 0.46875 → θ = cos⁻¹(0.46875) ≈ 62.0°

✔ Answer: ≈ 62.0°

---

6.) Triangle: Sides 21 and 29, right triangle

Assuming:
- One leg = 21
- Hypotenuse = 29
- Find angle opposite 21

sin(θ) = 21/29 ≈ 0.7241 → θ = sin⁻¹(0.7241) ≈ 46.4°

✔ Answer: ≈ 46.4°

---

7.) Triangle: Sides 16 and 29, right angle at bottom right, angle at bottom left is ?

Assuming:
- Adjacent to angle = 16
- Hypotenuse = 29
- So cos(θ) = 16/29 ≈ 0.5517 → θ = cos⁻¹(0.5517) ≈ 56.4°

✔ Answer: ≈ 56.4°

---

8.) Triangle: Sides 2 and 4, right triangle, angle at bottom right is ?

Assuming:
- Opposite = 2
- Adjacent = 4
- So tan(θ) = 2/4 = 0.5 → θ = tan⁻¹(0.5) ≈ 26.6°

✔ Answer: ≈ 26.6°

---

✔ Final Answers (Rounded to Nearest Degree or Decimal):

| Problem | Answer |
|--------|--------|
| 1.) | 20° |
| 2.) | 27.5° |
| 3.) | 25.0° |
| 4.) | 43.0° |
| 5.) | 62.0° |
| 6.) | 46.4° |
| 7.) | 56.4° |
| 8.) | 26.6° |

> 🔍 Note: These answers assume standard configurations where the unknown angle is opposite or adjacent to the given side. Double-check the diagrams for exact labeling (e.g., which side is opposite/adjacent).

---

If you can describe the exact labels in any problem (e.g., "In problem 1, side 24 is opposite the angle, 66 is adjacent"), I can refine the answer further.

Let me know!