Math Problem Statement

Consider a triangle ABC like the one below. Suppose that B = 129°, a = 36, and c = 29. Solve the triangle.

Solution

The problem involves solving a triangle using the given parameters:

  • B=129B = 129^\circ (angle at vertex B)
  • a=36a = 36 (length of side opposite angle AA)
  • c=29c = 29 (length of side opposite angle CC)

Step 1: Use the Law of Cosines

To find the unknown side bb (the side opposite angle BB), we can apply the Law of Cosines:

b2=a2+c22accos(B)b^2 = a^2 + c^2 - 2ac \cdot \cos(B)

Substitute the given values:

b2=362+29223629cos(129)b^2 = 36^2 + 29^2 - 2 \cdot 36 \cdot 29 \cdot \cos(129^\circ)

We'll calculate the value of bb.

Step 2: Find Angle AA

Next, we can use the Law of Sines to find angle AA. The Law of Sines is:

asin(A)=bsin(B)=csin(C)\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}

Since we know aa, bb, and BB, we can solve for AA.

36sin(A)=bsin(129)\frac{36}{\sin(A)} = \frac{b}{\sin(129^\circ)}

Solve for AA.

Step 3: Find Angle CC

Finally, since the sum of angles in a triangle is 180180^\circ, we can find angle CC as:

C=180ABC = 180^\circ - A - B

I'll compute these values for you now.The solution for the triangle is as follows:

  • Angle A28.4A \approx 28.4^\circ
  • Angle C22.6C \approx 22.6^\circ
  • Side b58.7b \approx 58.7

If you have any questions or would like additional details, let me know!

Here are some related questions to expand your understanding:

  1. How would the Law of Sines apply if angle AA was known instead of BB?
  2. Can there be more than one triangle satisfying the given conditions? Why or why not?
  3. How would you modify the Law of Cosines for a right-angled triangle?
  4. What would happen if angle BB were less than 9090^\circ?
  5. How does the accuracy of angle and side measurement affect solving triangles?

Tip: Always check if the angle you're calculating in the Law of Sines is valid by considering the range of possible values (0 to 180 degrees).

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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 - 2ac * cos(B)
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)

Theorems

Law of Cosines
Law of Sines

Suitable Grade Level

Grades 10-12