Problem 8 Solve the triangle \(\triangle A... [FREE SOLUTION] (2024)

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The Law of Sines is a fundamental tool in trigonometry for solving triangles. It relates the lengths of the sides of a triangle to the sines of its angles. It's especially useful in non-right triangles. The Law of Sines states that:
\(\frac{a}{\text{sin}(A)} = \frac{b}{\text{sin}(B)} = \frac{c}{\text{sin}(C)}\)
Where \(a, b, c\) are the lengths of the sides opposite the angles \(A, B, C\), respectively.

• Step 1: Identify and substitute known values into the Law of Sines.
• Step 2: Rearrange and solve for the unknowns.

For example, in the given exercise, we use: \(
\frac{15}{\text{sin}(94^\text{°})} = \frac{12}{\text{sin}(B)}\)
Notice how the sine of an angle works perfectly with side-length ratios. This concept is vital for accurately solving any triangle when at least one side length and the opposite angle are known.

One of the simplest yet powerful facts in trigonometry is that the sum of the angles in any triangle is always 180 degrees. This fact is known as the Triangle Angle Sum Property.

• Step 1: Identify any two angles of the triangle.
• Step 2: Subtract their sum from 180° to find the third angle.

For example, in our exercise: \(C = 180^\text{°} - 94^\text{°} - 53.13^\text{°} \approx 32.87^\text{°}\). By using this property, you can easily solve for the missing angle, which is essential to proceed with other calculations for any triangle.

Trigonometry forms the foundation for solving problems involving triangles. Some of the crucial elements include: sine, cosine, and tangent functions.

• Sine (sin): Opposite side over hypotenuse
• Cosine (cos): Adjacent side over hypotenuse
• Tangent (tan): Opposite side over adjacent side

Specific to our problem, the sine function is heavily used. For example, to solve for \(B\): \( \text{sin}(B) = \frac{12 \times \text{sin}(94^\text{°})}{15} \approx 0.8 \) Using the inverse sine function: \(\ B = \text{sin}^{-1}(0.8) \approx 53.13^\text{°} \) Understanding these trigonometric basics allows you to manipulate and understand the relationships between triangle's sides and angles effectively.