Solutions of Triangles
Key Concepts:
- Sine Rule: In any triangle ABC, a/sin A = b/sin B = c/sin C.
- Cosine Rule: In any triangle ABC, a² = b² + c² - 2bc cos A (and similar formulas for b² and c²).
- Tangent Rule: tan(A/2) = √[(s-b)(s-c)/(s(s-a))], where s = (a+b+c)/2.
- Area of a Triangle:
- Using sides: Δ = √[s(s-a)(s-b)(s-c)] (Heron's formula)
- Using base and height: Δ = (1/2) * base * height
- Using two sides and included angle: Δ = (1/2) * ab * sin C
Important Properties:
- The sum of the angles in a triangle is always 180°.
- The length of any side of a triangle is less than the sum of the other two sides.
- In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of
the other two sides (Pythagorean theorem).
Applications:
- Solving triangles given various combinations of sides and angles.
- Finding areas of triangles.
- Proving trigonometric identities.
- Solving problems in geometry, physics, and engineering.
Example:
Given a triangle ABC with a = 5, b = 7, and ∠C = 60°, find the remaining sides and angles.
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Find angle A: Using the cosine rule, a² = b² + c² - 2bc cos A.
- 5² = 7² + c² - 27c*cos(60°)
- 25 = 49 + c² - 7c
- c² - 7c + 24 = 0
- (c - 3)(c - 8) = 0
- So, c = 3 or c = 8. Since c cannot be less than a or b, c = 8.
-
Find angles B and C: Using the sine rule, a/sin A = b/sin B.
- 5/sin A = 7/sin B
- sin B = (7/5) * sin A
- Using a calculator, we find sin B ≈ 0.9325.
- So, B ≈ 68.75°.
- Finally, C = 180° - A - B ≈ 180° - 41.25° - 68.75° = 70°.