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Law of sines - Trigonometric Ratios and Angle Measures

Law of sines

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Notes:
Law of Sine
For any \triangle ABC,

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

Use the Law of Sine when given a pair!

Ambiguous case
Ambiguous case of the Law of Sine arises when given SSA (side-side-angle)

Step 1) Use the given angle to find the height of the triangle: h=bsin(A)h=b \sin (A)

Step 2) Check if,
SideaSide\;a < hh,
then no triangles
Sidea=hSide\;a=h,
then 1 triangle
SideaSide\;a > hh,
then 1 triangle
hh < SideaSide\;a < SidebSide\;b,
then 2 triangles

Step 3) Solve the triangle(s)!
  • 2.
    Given the following triangle ABC\triangle ABC,

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Law of sines

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