Math Problem Statement

Solve the triangle given: a = 8.999 in, c = 6.194 in, B = 75.72°. Find the length of side b, the measure of angle A, and the measure of angle C.

Solution

To solve this triangle, we'll use the given information:

  • Side a=8.999a = 8.999 inches,
  • Side c=6.194c = 6.194 inches,
  • Angle B=75.72B = 75.72^\circ.

The unknowns we need to find are:

  1. The length of side bb,
  2. The measure of angle AA,
  3. The measure of angle CC.

Step-by-Step Solution

  1. Finding side bb: We can apply the Law of Cosines: b2=a2+c22accos(B)b^2 = a^2 + c^2 - 2 \cdot a \cdot c \cdot \cos(B) Substitute a=8.999a = 8.999, c=6.194c = 6.194, and B=75.72B = 75.72^\circ.

  2. Finding angle AA: Once bb is found, we can use the Law of Sines: sinAa=sinBb\frac{\sin A}{a} = \frac{\sin B}{b} Solve for AA using the calculated value of bb.

  3. Finding angle CC: Using the fact that the sum of angles in a triangle is 180180^\circ: C=180BAC = 180^\circ - B - A

I'll calculate these values.The solutions are as follows:

  1. The length of side bb: 9.5849.584 inches (rounded to the nearest thousandth),
  2. The measure of angle AA: 65.5065.50^\circ (rounded to the nearest hundredth),
  3. The measure of angle CC: 38.7838.78^\circ (rounded to the nearest hundredth).

Would you like more detailed steps for any part of the solution, or have any questions?

Here are some related questions for further exploration:

  1. How would the solution change if angle BB were obtuse?
  2. What other trigonometric rules could solve this triangle?
  3. How would you solve it if only two sides and an angle were given?
  4. Can the Law of Sines give multiple solutions?
  5. What if side cc were shorter than side aa?

Tip: Remember to check the range of trigonometric values when using inverse functions to avoid ambiguous cases in triangles.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Law of Cosines
Law of Sines

Formulas

Law of Cosines: b^2 = a^2 + c^2 - 2 * a * c * cos(B)
Law of Sines: sin(A) / a = sin(B) / b
Sum of Angles in a Triangle: C = 180° - B - A

Theorems

Law of Cosines
Law of Sines

Suitable Grade Level

Grades 10-12