Identify if the illustration shows angle of | StudyX - Free Printable

Let's go through each of the 10 illustrations and determine whether they show an angle of elevation or an angle of depression, and then solve any problems that involve finding angles or trigonometric values.

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🔍 Definitions:

- Angle of Elevation: The angle formed when looking upward from a horizontal line to an object.
- Angle of Depression: The angle formed when looking downward from a horizontal line to an object.

These are always measured from the horizontal.

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✔ Problem Analysis & Solutions

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#### 1.
- Illustration: A person on the ground looks up at a crane (30 ft high), with a 70° angle between the ground and the line of sight.
- Answer: This is an angle of elevation because the person is looking up at the crane.
- ✔ Angle of Elevation

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#### 2.
- Illustration: A boat sees a lighthouse at a 12° angle above the horizontal.
- Answer: The boat is looking up at the lighthouse → angle of elevation.
- ✔ Angle of Elevation

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#### 3.
- Illustration: Right triangle with legs 300 and 400, angle x° at point A.
- Answer: The plane is above the ground, and we’re measuring from the ground (point A) upward → this is angle of elevation.
- ✔ Angle of Elevation

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#### 4.
- Illustration: A boat is pulling something underwater with a rope forming a 39° angle with the horizontal.
- Answer: The angle is measured from the horizontal downward to the rope → angle of depression.
- ✔ Angle of Depression

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#### 5. Angle P
- Triangle: Right triangle with right angle at R. Sides:
- PR = q (vertical leg)
- QR = p (horizontal leg)
- Hypotenuse PQ = r
- Angle P: At vertex P, between sides p and r.
- Since angle P is adjacent to side p and opposite to q, we can find it using trig functions.
- But no numerical values are given. So likely just identifying the angle.
- Answer: We are to identify angle P, which is one of the acute angles in the triangle.
- ✔ No numerical answer needed unless more info is provided. Just label: ∠P

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#### 6. sin C
- Triangle ABC, right-angled at B.
- AB = 28 (opposite to ∠C)
- BC = 21 (adjacent to ∠C)
- AC = 35 (hypotenuse)
- Use definition of sine:
$$
\sin C = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{AB}{AC} = \frac{28}{35} = \frac{4}{5} = 0.8
$$
- ✔ sin C = 0.8

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#### 7.
- Illustration: A person standing at a distance looks up at a tree.
- Answer: Person is looking up at the top of the tree → angle of elevation.
- ✔ Angle of Elevation

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#### 8. x°
- Right triangle with:
- One leg = 28
- Hypotenuse = 41
- Angle x° is opposite the side of length 28.
- Use sine:
$$
\sin x^\circ = \frac{28}{41}
$$
$$
x^\circ = \sin^{-1}\left(\frac{28}{41}\right) \approx \sin^{-1}(0.6829) \approx 42.7^\circ
$$
- ✔ x ≈ 42.7°

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#### 9.
- Illustration: A boat sees a lighthouse on a cliff.
- The angle is measured from the boat’s horizontal line up to the lighthouse.
- Answer: Looking up → angle of elevation.
- ✔ Angle of Elevation

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#### 10.
- Illustration: A person looks at the top of a tree, but the angle is drawn from the person’s eye level downward? Wait — actually, it shows the person looking up at the tree.
- The angle is at the person’s eye level, going up toward the tree.
- So, this is an angle of elevation.
- ✔ Angle of Elevation

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✔ Final Answers Summary:

| Number | Type of Angle | Notes |
|--------|---------------------------|-------|
| 1 | Angle of Elevation | Looking up at crane |
| 2 | Angle of Elevation | Boat sees lighthouse above |
| 3 | Angle of Elevation | From ground to airplane |
| 4 | Angle of Depression | Boat pulls down into water |
| 5 | ∠P | Label only; no value given |
| 6 | sin C = 0.8 | Calculated from triangle |
| 7 | Angle of Elevation | Looking up at tree |
| 8 | x ≈ 42.7° | Using inverse sine |
| 9 | Angle of Elevation | Boat sees lighthouse above |
| 10 | Angle of Elevation | Person looks up at tree |

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📌 Key Takeaways:

- Elevation: Look up → angle above horizontal.
- Depression: Look down → angle below horizontal.
- Always check from whose perspective the angle is measured.

Let me know if you'd like diagrams or step-by-step trigonometry for any of these!