Math Problem Statement

Solve for missing sides and angles, Given: C = 6.8 degrees B = 1.9 degrees c = 27

Solution

To solve the triangle, we will determine the missing sides and angles using the given data. Since two angles CC and BB are known, we can find the third angle AA, and then use the Law of Sines to find the missing sides aa and bb.

Step 1: Find angle AA

The sum of the angles in a triangle is 180180^\circ. Therefore: A=180CBA = 180^\circ - C - B A=1806.81.9=171.3A = 180^\circ - 6.8^\circ - 1.9^\circ = 171.3^\circ

Step 2: Apply the Law of Sines

The Law of Sines states: asinA=bsinB=csinC\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} We know c=27c = 27 and sinC\sin C. First, calculate sinC\sin C and sinB\sin B.

  • sinC=sin(6.8)\sin C = \sin(6.8^\circ)
  • sinB=sin(1.9)\sin B = \sin(1.9^\circ)

Then: asinA=27sin(6.8)andbsinB=27sin(6.8)\frac{a}{\sin A} = \frac{27}{\sin(6.8^\circ)} \quad \text{and} \quad \frac{b}{\sin B} = \frac{27}{\sin(6.8^\circ)}

I’ll calculate these values. math

Given values

C = 6.8 # in degrees B = 1.9 # in degrees c = 27 # side c

Convert angles to radians for sine calculation

C_rad = math.radians(C) B_rad = math.radians(B)

Find A in degrees

A = 180 - C - B

Convert A to radians

A_rad = math.radians(A)

Calculate sines of angles

sin_C = math.sin(C_rad) sin_B = math.sin(B_rad) sin_A = math.sin(A_rad)

Calculate sides using the Law of Sines

a = (c * sin_A) / sin_C b = (c * sin_B) / sin_C

a, b, AThe missing sides and angle are as follows:

  • A171.3A \approx 171.3^\circ
  • a34.49a \approx 34.49
  • b7.56b \approx 7.56

These values were calculated using the Law of Sines. Let me know if you would like detailed steps or have further questions.

Here are 5 related questions for further exploration:

  1. How does the Law of Sines apply to obtuse triangles?
  2. What would change if the triangle were a right triangle?
  3. How can the Law of Cosines be used in this situation?
  4. Why does the triangle's shape depend on the sum of angles equaling 180180^\circ?
  5. Can you confirm the side lengths satisfy the triangle inequality?

Tip: Always check the feasibility of triangle side lengths and angles to ensure no input errors.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Triangle Properties
Law of Sines

Formulas

Sum of angles in a triangle: A + B + C = 180°
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)

Theorems

Law of Sines

Suitable Grade Level

Grades 10-12