Math Problem Statement

Solve the following right triangle: B = 17.9°, a = 12.6. Find the missing side b, hypotenuse c, and angle A.

Solution

To solve the right triangle, the given data includes:

  1. Angle B=17.9B = 17.9^\circ
  2. Side a=12.6a = 12.6 (the side opposite angle AA).

Goal:

Find:

  1. The hypotenuse (cc).
  2. The other leg (bb).
  3. The remaining angle (AA).

Solution:

  1. Find the remaining angle AA: In a right triangle, the sum of the angles is 90+A+B=18090^\circ + A + B = 180^\circ. Therefore: A=90B=9017.9=72.1A = 90^\circ - B = 90^\circ - 17.9^\circ = 72.1^\circ

  2. Find the hypotenuse cc: Using the sine function: sin(B)=oppositehypotenusesoc=asin(B).\sin(B) = \frac{\text{opposite}}{\text{hypotenuse}} \quad \text{so} \quad c = \frac{a}{\sin(B)}. Substituting: c=12.6sin(17.9)=12.60.30741.04.c = \frac{12.6}{\sin(17.9^\circ)} = \frac{12.6}{0.307} \approx 41.04.

  3. Find the other leg bb: Using the tangent function: tan(B)=oppositeadjacentsob=atan(B).\tan(B) = \frac{\text{opposite}}{\text{adjacent}} \quad \text{so} \quad b = \frac{a}{\tan(B)}. Substituting: b=12.6tan(17.9)=12.60.32438.89.b = \frac{12.6}{\tan(17.9^\circ)} = \frac{12.6}{0.324} \approx 38.89.

Final Answer:

  • A=72.1A = 72.1^\circ
  • c41.04c \approx 41.04
  • b38.89b \approx 38.89

Would you like further explanations on any of these steps?

Here are 5 questions you could explore:

  1. What are the primary trigonometric ratios, and how are they applied in right triangles?
  2. Can this method work for non-right triangles? Why or why not?
  3. How would the calculations change if side bb was given instead of side aa?
  4. What tools can simplify solving trigonometric problems in real life?
  5. How do you verify that your solutions are correct in problems like these?

Tip: When solving for a triangle's missing sides or angles, always confirm that the relationships between the sides and angles satisfy the Pythagorean theorem and the sum of angles rule!

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangles
Angle Sum Property

Formulas

A = 90° - B
c = a / sin(B)
b = a / tan(B)

Theorems

Angle Sum Property in Triangles
Trigonometric Ratios: Sine and Tangent

Suitable Grade Level

Grades 9-12