Question

Please help with the law of cosines

Answer 1

38.2°

Explanation:

To find the measure of the indicated angle in the given triangle, we can use the Law of Cosines.

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

In this case, as the side measuring 5 units is opposite the unknown angle:

• 
  `a = 7`
• 
  `b = 8`
• 
  `c = 5`

Substitute these values into the formula:

`5^2=7^2+8^2-2(7)(8)\cos C`

Now, solve for angle C:

`25=49+64-112\cos C`

`25=113-112\cos C`

`112\cos C=113-25`

`112\cos C=88`

`\cos C=(88)/(112)=(11)/(14)`

`C=\cos^(-1)\left((11)/(14)\right)`

`C=38.2132107...`

`C=38.2^(\circ)\; \sf (nearest\;tenth)`

Therefore, the measure of the indicated angle is 38.2°, rounded to the nearest tenth of a degree.

Answer 2

`\sf m \angle C \approx 38.2^\circ`

Explanation:

Let the triangle be ABC where

AB = c = 5

BC = a = 8

AC = b = 7

To find:

`\sf \angle C = ?` Using Law of cosine

Solution:

The Law of Cosines is a trigonometric formula used to find the measure of an angle in a triangle when the lengths of all three sides are known. The formula is given by:

`\sf c^2 = a^2 + b^2 - 2ab \cos(C)`

where:

-
`\sf c` is the length of the side opposite angle
`\sf C`,

-
`\sf a` and
`\sf b` are the lengths of the other two sides.

In this case, we want to find angle
`\sf C`, and we know the lengths of the sides opposite angles
`\sf A`,
`\sf B`, and
`\sf C` as
`\sf a = BC = 8`,
`\sf b = AC = 7`, and
`\sf c = AB = 5`.

The Law of Cosines formula becomes:

`\sf 5^2 = 7^2 + 8^2 - 2 \cdot 7 \cdot 8 \cdot \cos(C)`

Solving for
`\sf \cos(C)`:

`\sf 25 = 49 + 64 - 112 \cos(C)`

Combine like terms:

`\sf -88 = -112 \cos(C)`

Now, solve for
`\sf \cos(C)`:

`\begin{aligned} \sf \cos(C) & = (-88)/(-112) \\\\ & = (22)/(28) \\\\ & = (11)/(14) \end{aligned}`

Now, find the angle
`\sf C` by taking the arccosine (inverse cosine) of
`\sf (11)/(14)`:

`\sf C = \cos^(-1)\left((11)/(14)\right)`

Using a calculator to find the numerical value of
`\sf C`.

`\sf C = 38.2132107`

The result is approximately
`\sf 38.2^\circ` in nearest tenth

So,
`\sf m \angle C \approx 38.2^\circ`