Question

If the measure of A is 55°, b = 12 and c = 7 find angle C.

Answer 1

C = 35.7°

Explanation:

In the case of triangle ABC, we are given the measures of two sides (b and c) as well as the included angle A.

To determine the measure of angle C, we first need to find the length of side a by using the Law of Cosines. Once we have found the length of side a, we can then use the Law of Sines to find the measure of angle C.

`\boxed{\begin{array}{l}\underline{\textsf{Law of Cosines}}\\\\a^2=b^2+c^2-2bc \cos A\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a, b$ and $c$ are the sides.}\\\phantom{ww}\bullet\;\textsf{$A$ is the angle opposite side $a$.}\end{array}}`

Substitute A = 55°, b = 12 and c = 7 into the Law of Cosines formula, then solve for the exact value of a:

`a^2=12^2+7^2-2(12)(7)\cos (55^(\circ))`

`a^2=144+49-168\cos (55^(\circ))`

`a^2=193-168\cos (55^(\circ))`

`a=\sqrt{193-168\cos (55^(\circ))}`

Now, we have the length of all three sides (a, b and c) and the measure of angle A, we can use the Law of Sines to find the measure of angle C.

`\boxed{\begin{array}{l}\underline{\textsf{Law of Sines}} \\\\(\sin A)/(a)=(\sin B)/(b)=(\sin C)/(c)\\\\\textsf{where:}\\\phantom{ww}\bullet \;\textsf{$A, B$ and $C$ are the angles.}\\\phantom{ww}\bullet\;\textsf{$a, b$ and $c$ are the sides opposite the angles.}\end{array}}`

Substitute the values of a, c, and A into the formula:

`\begin{aligned}(\sin A)/(a)&=(\sin C)/(c)\\\\\frac{\sin 55^(\circ)}{\sqrt{193-168\cos (55^(\circ))}}&=(\sin C)/(7)\end{aligned}`

Now, solve for C:

`\begin{aligned}\sin C&=\frac{7\sin 55^(\circ)}{\sqrt{193-168\cos (55^(\circ))}}\\\\C&=\sin^(-1)\left(\frac{7\sin 55^(\circ)}{\sqrt{193-168\cos (55^(\circ))}}\right)\\\\C&=35.6824161...^(\circ)\\\\C&=35.7^(\circ)\; \sf (nearest\;tenth)\end{aligned}`

Therefore, the measure of angle C is:

`\Large\boxed{\boxed{C=35.7^(\circ)}}`

Answer 2

`\sf m\angle C \approx 35.7^\circ`

Explanation:

Given:

• Measure of angle A = 55°
• Side
  `\sf b = 12`
• Side
  `\sf c = 7`

To find:

• 
  `\sf m\angle C = ?`

Solution:

To find angle C first we need to find a, and we can find angle C after that:

We can find a by using Law of cosines:

The Law of Cosines states that:

`\sf a^2 = b^2 + c^2 - 2bc \cos(A)`

Substituting the given values into the equation:

`\sf a^2 = 12^2 + 7^2 - 2(12)(7) \cos(55^\circ)`

`\sf a^2 = 144+49 - 168 \cos(55^\circ)`

`\sf a^2 = 193 - 168\cdot 0.5735764364`

`\sf a^2 = 193 - 96.36084131`

`\sf a^2 = 96.63915869`

`\sf a = √(96.63915869)`

`\sf a \approx 9.8305217913394`

Now

Find angle
`\sf C` using the Law of Sines

The Law of Sines states that:

`\sf (a)/(\sin(A)) = (b)/(\sin(B)) = (c)/(\sin(C))`

We want to find the angle
`\sf C`, so rearrange the equation:

`\sf \sin(C) = (c)/(a) \sin(A)`

Substituting the given values:

`\sf \sin(C) = (7)/(9.8305217913394) \sin(55^\circ)`

`\sf \sin(C) = 0.7120680009241 \sin(55^\circ)`

`\sf \sin(C) = 0.7120680009241 \cdot × 0.8191520443`

`\sf \sin(C) = 0.58329195863759080413763`

Taking the inverse sine:

`\sf m \angle C =sin^(-1) (0.58329195863759080413763)`

`\sf m \angle C = (35.682416150506 )^\circ`

`\sf m\angle C \approx 35.7^\circ \textsf{(in nearest tenth)}`

So, the measure of angle C is 35.7 degree in nearest tenth.