The solution of triangles refers to determining the unknown sides, angles, or other properties of a triangle using given information. This typically involves applying geometric principles and trigonometric laws. Here are the main methods used:

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1. Using the Pythagorean Theorem

• Applicable for right triangles only.
• Formula: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.

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2. Using Trigonometric Ratios (Right Triangles)

For a right triangle:

• $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$
• $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
• $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

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3. Law of Sines

• Applicable for any triangle.
• Formula:
  $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ Here, $a, b, c$ are the sides opposite angles $A, B, C$, respectively.

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4. Law of Cosines

• Applicable for any triangle, especially when the Law of Sines cannot be used.
• Formula:
  $c^2 = a^2 + b^2 - 2ab\cos C$ (and analogous formulas for other sides).

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5. Area of a Triangle

Using Base and Height:

• $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

Using Trigonometry:

• $\text{Area} = \frac{1}{2}ab\sin C$

Using Heron's Formula:

• When all sides are known: $s = \frac{a + b + c}{2}$ $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$

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Problem Types in Triangle Solutions:

1. Given Two Angles and One Side (AAS or ASA):

   • Use the Law of Sines to solve for unknowns.

2. Given Two Sides and One Included Angle (SAS):

   • Use the Law of Cosines to find the third side, then Law of Sines or angles.

3. Given Three Sides (SSS):

   • Use the Law of Cosines to determine one angle, then proceed with Law of Sines.

4. Given Two Sides and One Non-Included Angle (SSA):

   • Apply the Law of Sines. This may lead to two solutions (ambiguous case).

5. Right Triangle Problems:

   • Use trigonometric ratios or Pythagorean theorem.

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Would you like to explore any specific type of problem in detail?

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5 Questions for Exploration:

1. How does the Law of Sines apply when solving a triangle with two angles known?
2. Can the Law of Cosines determine all angles in a triangle if sides are known?
3. What are the steps to handle the ambiguous case in SSA triangle problems?
4. How does Heron’s formula relate to other area calculation methods for triangles?
5. In what scenarios can trigonometric ratios replace the Law of Sines or Cosines?

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Tip:

Always sketch the triangle and label given values before starting any calculations. This reduces errors significantly.