Question

NO LINKS!! URGENT HELP PLEASE!!

Please help me with #31 & 32​

Answer 1

31. m∠E = 56.1°

32. c = 24.9 inches

Explanation:

Question 31

The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles. It states that the ratio of the length of a side of a triangle to the sine of the opposite angle is constant for all three sides of the triangle.

`\boxed{\begin{minipage}{7.6 cm}\underline{Law of Sines} \\\\$(\sin A)/(a)=(\sin B)/(b)=(\sin C)/(c)$\\\\\\where:\\ \phantom{ww}$\bullet$ $A, B$ and $C$ are the angles. \\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides opposite the angles.\\\end{minipage}}`

Given values of triangle DEF:

• m∠D = 81°
• d = 25
• e = 21

To find m∠E, substitute the values into the Law of Sines formula and solve for E:

`(\sin D)/(d)=(\sin E)/(e)`

`(\sin 81^(\circ))/(25)=(\sin E)/(21)`

`\sin E=(21\sin 81^(\circ))/(25)`

`E=\sin^(-1)\left((21\sin 81^(\circ))/(25)\right)`

`E=56.1^(\circ)\; \sf(nearest\;tenth)`

Therefore, the measure of angle E is 56.1°, to the nearest tenth.

See the attachment for the accurate drawing of triangle DEF.

`\hrulefill`

Question 32

The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.

`\boxed{\begin{minipage}{6 cm}\underline{Law of Cosines} \\\\$c^2=a^2+b^2-2ab \cos C$\\\\where:\\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides.\\ \phantom{ww}$\bullet$ $C$ is the angle opposite side $c$. \\\end{minipage}}`

From inspection of triangle ABC:

• C = 125°
• a = 13 inches
• b = 15 inches

To find the length of side c, substitute the values into the Law of Cosines formula and solve for c:

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

`c^2=13^2+15^2-2(13)(15) \cos 125^(\circ)`

`c^2=169+225-390 \cos 125^(\circ)`

`c^2=394-390 \cos 125^(\circ)`

`c=\sqrt{394-390 \cos 125^(\circ)}`

`c=24.8534667...`

`c=24.9\; \sf inches\;(nearest\;tenth)`

Therefore, the length of side c is 24.9 inches, to the nearest tenth.