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Sagot :
Certainly! We'll solve the given problem by systematically working through the steps.
1. Understand the Problem:
- The angles of the triangle are in Arithmetic Progression (AP).
- The greatest angle is double the least angle.
- The sum of the angles of any triangle is always [tex]\(180^\circ\)[/tex].
2. Representation:
- Let the smallest angle be [tex]\(a\)[/tex].
- Since the angles are in AP, we can represent the other two angles as [tex]\(a + d\)[/tex] and [tex]\(a + 2d\)[/tex], where [tex]\(d\)[/tex] is the common difference in the AP.
3. Given Condition:
- The greatest angle is double the least angle:
[tex]\[ a + 2d = 2a \][/tex]
- Simplifying this, we get:
[tex]\[ 2d = a \implies a = 2d \][/tex]
4. Sum of Angles:
- The sum of the angles of the triangle must be [tex]\(180^\circ\)[/tex]:
[tex]\[ a + (a + d) + (a + 2d) = 180 \][/tex]
- Substitute [tex]\(a = 2d\)[/tex]:
[tex]\[ 2d + (2d + d) + (2d + 2d) = 180 \][/tex]
- Simplify the equation:
[tex]\[ 2d + 3d + 4d = 180 \][/tex]
[tex]\[ 9d = 180 \][/tex]
- Solve for [tex]\(d\)[/tex]:
[tex]\[ d = 20 \][/tex]
5. Determine the Angles:
- Smallest angle ([tex]\(a\)[/tex]):
[tex]\[ a = 2d = 2 \times 20 = 40^\circ \][/tex]
- Middle angle ([tex]\(a + d\)[/tex]):
[tex]\[ 40^\circ + 20^\circ = 60^\circ \][/tex]
- Largest angle ([tex]\(a + 2d\)[/tex]):
[tex]\[ 40^\circ + 40^\circ = 80^\circ \][/tex]
Therefore, the angles in degrees are:
[tex]\[ 40^\circ, 60^\circ, 80^\circ \][/tex]
6. Convert the Angles to Radians:
- Use the conversion formula from degrees to radians:
[tex]\[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \][/tex]
- For [tex]\(40^\circ\)[/tex]:
[tex]\[ 40^\circ \times \frac{\pi}{180} \approx 0.6981 \: \text{radians} \][/tex]
- For [tex]\(60^\circ\)[/tex]:
[tex]\[ 60^\circ \times \frac{\pi}{180} \approx 1.0472 \: \text{radians} \][/tex]
- For [tex]\(80^\circ\)[/tex]:
[tex]\[ 80^\circ \times \frac{\pi}{180} \approx 1.3963 \: \text{radians} \][/tex]
Therefore, the angles in radians are approximately:
[tex]\[ 0.6981 \: \text{radians}, 1.0472 \: \text{radians}, 1.3963 \: \text{radians} \][/tex]
In summary, the angles of the triangle are:
- In degrees: [tex]\(40^\circ, 60^\circ, 80^\circ\)[/tex]
- In radians: [tex]\(0.6981\)[/tex] radians, [tex]\(1.0472\)[/tex] radians, [tex]\(1.3963\)[/tex] radians
1. Understand the Problem:
- The angles of the triangle are in Arithmetic Progression (AP).
- The greatest angle is double the least angle.
- The sum of the angles of any triangle is always [tex]\(180^\circ\)[/tex].
2. Representation:
- Let the smallest angle be [tex]\(a\)[/tex].
- Since the angles are in AP, we can represent the other two angles as [tex]\(a + d\)[/tex] and [tex]\(a + 2d\)[/tex], where [tex]\(d\)[/tex] is the common difference in the AP.
3. Given Condition:
- The greatest angle is double the least angle:
[tex]\[ a + 2d = 2a \][/tex]
- Simplifying this, we get:
[tex]\[ 2d = a \implies a = 2d \][/tex]
4. Sum of Angles:
- The sum of the angles of the triangle must be [tex]\(180^\circ\)[/tex]:
[tex]\[ a + (a + d) + (a + 2d) = 180 \][/tex]
- Substitute [tex]\(a = 2d\)[/tex]:
[tex]\[ 2d + (2d + d) + (2d + 2d) = 180 \][/tex]
- Simplify the equation:
[tex]\[ 2d + 3d + 4d = 180 \][/tex]
[tex]\[ 9d = 180 \][/tex]
- Solve for [tex]\(d\)[/tex]:
[tex]\[ d = 20 \][/tex]
5. Determine the Angles:
- Smallest angle ([tex]\(a\)[/tex]):
[tex]\[ a = 2d = 2 \times 20 = 40^\circ \][/tex]
- Middle angle ([tex]\(a + d\)[/tex]):
[tex]\[ 40^\circ + 20^\circ = 60^\circ \][/tex]
- Largest angle ([tex]\(a + 2d\)[/tex]):
[tex]\[ 40^\circ + 40^\circ = 80^\circ \][/tex]
Therefore, the angles in degrees are:
[tex]\[ 40^\circ, 60^\circ, 80^\circ \][/tex]
6. Convert the Angles to Radians:
- Use the conversion formula from degrees to radians:
[tex]\[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \][/tex]
- For [tex]\(40^\circ\)[/tex]:
[tex]\[ 40^\circ \times \frac{\pi}{180} \approx 0.6981 \: \text{radians} \][/tex]
- For [tex]\(60^\circ\)[/tex]:
[tex]\[ 60^\circ \times \frac{\pi}{180} \approx 1.0472 \: \text{radians} \][/tex]
- For [tex]\(80^\circ\)[/tex]:
[tex]\[ 80^\circ \times \frac{\pi}{180} \approx 1.3963 \: \text{radians} \][/tex]
Therefore, the angles in radians are approximately:
[tex]\[ 0.6981 \: \text{radians}, 1.0472 \: \text{radians}, 1.3963 \: \text{radians} \][/tex]
In summary, the angles of the triangle are:
- In degrees: [tex]\(40^\circ, 60^\circ, 80^\circ\)[/tex]
- In radians: [tex]\(0.6981\)[/tex] radians, [tex]\(1.0472\)[/tex] radians, [tex]\(1.3963\)[/tex] radians
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