MathIsimple – Simple, Friendly Math Tools & Learning

1

Law of Sines Basic Application

Problem:

In $\triangle ABC$ , given $b = 2$ , $B = 30°$ , and $C = 45°$ , find $c$ .

2

Law of Sines Side-Angle Conversion

Problem:

In $\triangle ABC$ , let sides opposite to angles $A, B, C$ be $a, b, c$ . Given $\sin(B+C) + \sin A = \frac{3}{2}$ and $b = \sqrt{3}c$ , find angle $C$ .

3

Circumradius from Law of Cosines

Problem:

In $\triangle ABC$ , given $c = 1$ , $b = 2$ , and $A = 60°$ , find the circumradius $R$ of $\triangle ABC$ .

4

Triangle Shape Determination

Problem:

In $\triangle ABC$ , given $B = 2C$ and $b = 2a$ . Determine the type of triangle.

5

Triangle Area Formula

Problem:

In $\triangle ABC$ , given $B = 60°$ , $\sin A = 2\sin C$ , and $b = 2\sqrt{3}$ . Find the area of $\triangle ABC$ .

6

Maximum Perimeter Problem

Problem:

In $\triangle ABC$ , given $c\cos A + \sqrt{3}c\sin A - b + 2a = 0$ and $c = 3$ . Find the maximum perimeter of $\triangle ABC$ .

7

Maximum Area Problem

Problem:

In $\triangle ABC$ , given $a = 1$ and $(a+b)\sin A\sin B + (c-b)\sin C = 0$ . Find the maximum area $S$ of $\triangle ABC$ .

8

Quadrilateral Area Maximum

Problem:

In convex quadrilateral $ABCD$ , $AB \perp AD$ , $|AB| = |AD|$ , $BC = 4$ , $CD = 2$ , and $\cos\angle BCD = \frac{1}{4}$ . Find the maximum area of quadrilateral $ABCD$ .

9

Comprehensive Application

Problem:

In $\triangle ABC$ , given $a\cos B = \sqrt{3}$ and $b\sin A = 1$ .

(1) Find $\angle B$ .

(2) If $b = 2$ , find the area of $\triangle ABC$ .

10

Equilateral Triangle Verification

Problem:

In $\triangle ABC$ , sides $a, b, c$ are in arithmetic progression. The circle with diameter $AC$ has area $2\pi$ . If the area of $\triangle ABC$ is $2\sqrt{3}$ , determine the shape of the triangle.

Solving Triangles