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Question

In ABC c=11cm b=7cm<B=28°

-determine the measure of<C in each diagram

-find the measure of <A in each diagram

-calculate the length of BC in each diagram

Answer 1

Side c = 11 cm, Side b = 7 cm and Angle B = 28^o.

From the above triangle, sin(B) = opposite side/hypotenuse = h/c ⇒ h = c * sin(B) = 11 * sin(28) ≅ 5.16.

Necessary condition : h(=5.16) < b(=7) < c(=11), So the possible triangles are two.

By the Law of Sines in the reciprocal form

sin(C) / c = sin(B) / b

sin(C) = c * [sin(B) / b]

Substitute the values of B = 28, b = 7 and c = 11 in the above formula.

sin(C) = (11) * [sin(28) / 7] ≅ 0.7377.

C =  sin ^-1(0.7377) ≅ 47.54⁰.

There are two angles , C₁ = 47.54⁰ and C₂ = 180 - 47.54⁰ = 132.46⁰.

 

For C₁ = 47.54⁰, we obtain

A = 180 - B - C₁ = 180 - 28 - 47.54 = 104.46⁰.

The remaining side is

a/sin(A) = b/sin(B)

a = [b/sin(B)] * sin(A)

a = [7/sin(28)] * sin(104.46) ≅ 14.44 cm.

Angles A = 104.46⁰, C = 47.54⁰ and side a = 14.44 cm.

The resulting triangle is shown below.

For C₂ = 132.46⁰, we obtain

A = 180 - B - C₂ = 180 - 28 - 132.46 = 19.54⁰.

The remaining side is

a/sin(A) = b/sin(B)

a = [b/sin(B)] * sin(A)

a = [7/sin(28)] * sin(19.54) ≅ 5 cm.

Angles A = 19.54⁰, C = 132.46⁰ and side a = 5 cm.

The resulting triangle is shown below.