Answer:
a) m∠C = 76°, a = 7.84, b = 4.37
Step-by-step explanation:
In triangle ABC, the vertices are A, B and C, and the sides opposite the vertices are a, b and c.
The interior angles of a triangle sum to 180°. Therefore:
[tex]m\angle A + m\angle B+m\angle C=180^{\circ}[/tex]
Given that m∠A = 72° and m∠B = 32°, then:
[tex]\begin{aligned}72^{\circ} + 32^{\circ}+m\angle C&=180^{\circ}\\\\104^{\circ}+m\angle C&=180^{\circ}\\\\m\angle C&=180^{\circ}-104^{\circ}\\\\m\angle C&=76^{\circ}\end{aligned}[/tex]
Therefore, the measure of angle C is 76°.
To find the measures of the remaining sides, we can use the Law of Sines:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Law of Sines}} \\\\\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin 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}}[/tex]
In this case:
Substitute the values into the equation:
[tex]\dfrac{a}{\sin 72^{\circ}}=\dfrac{b}{\sin 32^{\circ}}=\dfrac{8}{\sin 76^{\circ}}[/tex]
Solve for side a:
[tex]\begin{aligned}\dfrac{a}{\sin 72^{\circ}}&=\dfrac{8}{\sin 76^{\circ}}\\\\a&=\dfrac{8\sin 72^{\circ}}{\sin 76^{\circ}}\\\\a&=7.8413744638...\\\\a&=7.84\; \sf (2 \;d.p.)\end{aligned}[/tex]
Solve for side b:
[tex]\begin{aligned}\dfrac{b}{\sin 32^{\circ}}&=\dfrac{8}{\sin 76^{\circ}}\\\\b&=\dfrac{8\sin 32^{\circ}}{\sin 76^{\circ}}\\\\b&=4.3691361293...\\\\b&=4.37\; \sf (2 \;d.p.)\end{aligned}[/tex]
Answer:
a : m∠C = 76°, a = 7.84, b = 4.37
Step-by-step explanation:
To find the measures of the remaining sides and angles in a triangle, we can use the Law of Sines and the fact that the sum of interior angles in a triangle is always 180°.
Given [tex] m\angle A = 72^\circ [/tex], [tex] m\angle B = 32^\circ [/tex], and side [tex] c = 8 [/tex], let's denote the other angle as [tex] \angle C [/tex] and the remaining sides as [tex] a [/tex] and [tex] b [/tex].
Since [tex] m\angle C [/tex] is the remaining angle, we can find it using the fact that the sum of interior angles in a triangle is 180°:
[tex] m\angle C = 180^\circ - m\angle A - m\angle B [/tex]
[tex] m\angle C = 180^\circ - 72^\circ - 32^\circ [/tex]
[tex] m\angle C = 76^\circ [/tex]
Now, we can use the Law of Sines to find the remaining sides. The Law of Sines states:
[tex] \dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} [/tex]
First, let's find a side [tex] a [/tex]:
[tex] \dfrac{a}{\sin A} = \dfrac{c}{\sin C} [/tex]
[tex] \dfrac{a}{\sin 72^\circ} = \dfrac{8}{\sin 76^\circ} [/tex]
Now, solve for [tex] a [/tex]:
[tex] a = \dfrac{\sin 72^\circ \times 8}{\sin 76^\circ} \\\\ = 7.841374464 \\\\ \approx 7.84 [/tex]
Next, let's find side [tex] b [/tex]:
[tex] \dfrac{b}{\sin B} = \dfrac{c}{\sin C} [/tex]
[tex] \dfrac{b}{\sin 32^\circ} = \dfrac{8}{\sin 76^\circ} [/tex]
Now, solve for [tex] b [/tex]:
[tex] b = \dfrac{\sin 32^\circ \times 8}{\sin 76^\circ} \\\\ = 4.369136129\\\\ \approx 4.37 [/tex]
So, the measure of remaining sides and angles are:
a : m∠C = 76°, a = 7.84, b = 4.37
