please help!

Question

In ABC c=11cm b=7cm<B=38°

-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 = 38^o.

From the above triangle, sin(B) = opposite side/hypotenuse = h/c ⇒ h = c * sin(B) = 11 * sin(38) ≅ 6.8.

Necessary condition : h(=6.8) < 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 = 38, b = 7 and c = 11 in the above formula.

sin(C) = (11) * [sin(38) / 7] ≅ 0.9675.

C =  sin ^-1(0.9675) ≅ 75.35⁰.

There are two angles , C₁ = 75.35⁰ and C₂ = 180 - 75.35⁰ = 104.65⁰.

 

For C₁ = 75.35⁰, we obtain

A = 180 - B - C₁ = 180 - 38 - 75.35 = 66.65⁰.

The remaining side is

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

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

a = [7/sin(38)] * sin(66.65) ≅ 10.44 cm.

Angles A = 66.65⁰, C = 75.35⁰ and side a = 10.44 cm.

The resulting triangle is shown below.

 

For C₂ = 104.65⁰, we obtain

A = 180 - B - C₂ = 180 - 38 - 104.65 = 37.35⁰.

The remaining side is

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

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

a = [7/sin(38)] * sin(37.35) ≅ 6.9 cm.

Angles A = 37.35⁰, C = 104.65⁰ and side a = 6.9 cm.

The resulting triangle is shown below.