Math Problem Statement

For the triangle, m∠C = 13°, m∠A = 22°, c = 9, find side a to the nearest whole number.

Solution

We are tasked with finding side aa of a triangle given the following:

  • C=13\angle C = 13^\circ,
  • A=22\angle A = 22^\circ,
  • Side c=9c = 9.

Solution:

  1. Find the third angle B\angle B: B=180AC\angle B = 180^\circ - \angle A - \angle C B=1802213=145\angle B = 180^\circ - 22^\circ - 13^\circ = 145^\circ

  2. Apply the Law of Sines: The Law of Sines states: asin(A)=bsin(B)=csin(C)\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} Using csin(C)\frac{c}{\sin(C)}, we solve for aa: a=csin(A)sin(C)a = c \cdot \frac{\sin(A)}{\sin(C)}

  3. Substitute the known values: a=9sin(22)sin(13)a = 9 \cdot \frac{\sin(22^\circ)}{\sin(13^\circ)}

  4. Calculate the sines and solve: Using a calculator: sin(22)0.3746,sin(13)0.2249\sin(22^\circ) \approx 0.3746, \quad \sin(13^\circ) \approx 0.2249 a=90.37460.224991.66514.985a = 9 \cdot \frac{0.3746}{0.2249} \approx 9 \cdot 1.665 \approx 14.985

  5. Round to the nearest whole number: a15a \approx 15

Final Answer:

The side aa is approximately 15.

Would you like a detailed explanation of any step or further questions?

Here are five related questions to consider:

  1. How is the Law of Sines used in non-right triangles?
  2. Why do we need to find the missing angle in a triangle before applying trigonometric laws?
  3. What other methods can be used to solve triangles (e.g., Law of Cosines)?
  4. Can the Law of Sines determine whether a triangle is obtuse, acute, or right?
  5. What happens when given a side and two angles (SAA) in a triangle?

Tip: Always ensure your calculator is set to degrees when solving angle-related problems.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Triangles
Law of Sines

Formulas

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)

Theorems

Law of Sines

Suitable Grade Level

Grades 10-12