Answer:
6) B = 45.79°, C = 74.21°, b = 7.45
7) B = 60.9°, b = 19.32, c = 6.36
8) B = 101.1°, a = 1.35, b = 3.23
Step-by-step explanation:
The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant, and this ratio is equal for all three sides.
[tex]\boxed{\begin{array}{l}\underline{\textsf{Law of Sines}} \\\\\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}\\\\\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]
To solve a triangle using the Law of Sines, substitute the given information into the appropriate equation derived from the Law of Sines to find the unknown angles and side lengths.
[tex]\hrulefill[/tex]
Question 6
Given:
Substitute the values of A, a and c into the equation and solve for angle C:
[tex]\dfrac{\sin 60^{\circ}}{9}=\dfrac{\sin C}{10}\\\\\\\sin C=\dfrac{10\sin 60^{\circ}}{9}\\\\\\C=\sin^{-1}\left(\dfrac{10\sin 60^{\circ}}{9}\right)\\\\\\C=74.20683095...^{\circ}\\\\\\C=74.21^{\circ}\;\sf (2\;d.p.)[/tex]
Angles in a triangle sum to 180°. Therefore:
[tex]B = 180^{\circ}-A-C\\\\B = 180^{\circ}-60^{\circ}-74.20683095...^{\circ}\\\\B=45.793169048...^{\circ}\\\\B=45.79^{\circ}\;\sf (2\;d.p.)[/tex]
Use the Law of Sines again to find the length of side b:
[tex]\dfrac{9}{\sin 60^{\circ}}=\dfrac{b}{\sin (45.793169048...^{\circ})}\\\\\\b=\dfrac{9\sin (45.793169048...^{\circ})}{\sin 60^{\circ}}\\\\\\b=7.44948974278...\\\\\\b=7.45\;\sf (2\;d.p.)[/tex]
[tex]\hrulefill[/tex]
Question 7
Given:
- A = 102.4°
- C = 16.7°
- a = 21.6
Angles in a triangle sum to 180°. Therefore:
[tex]B = 180^{\circ}-A-C\\\\B = 180^{\circ}-102.4^{\circ}-16.7^{\circ}\\\\B=60.9^{\circ}[/tex]
Substitute the values of A, B and a into the Law of Sines equation and solve for side length b:
[tex]\dfrac{21.6}{\sin 102.4^{\circ}}=\dfrac{b}{\sin 60.9^{\circ}}\\\\\\b=\dfrac{21.6\sin 60.9^{\circ}}{\sin 102.4^{\circ}}\\\\\\b=19.324271238...\\\\\\b=19.32\;\sf (2\;d.p.)[/tex]
Substitute the values of A, C and a into the Law of Sines equation and solve for side length c:
[tex]\dfrac{21.6}{\sin 102.4^{\circ}}=\dfrac{c}{\sin 16.7^{\circ}}\\\\\\c=\dfrac{21.6\sin 16.7^{\circ}}{\sin 102.4^{\circ}}\\\\\\c=6.35524051...\\\\\\c=6.36\;\sf (2\;d.p.)[/tex]
[tex]\hrulefill[/tex]
Question 8
Given:
- A = 24.3°
- C = 54.6°
- c = 2.68
Angles in a triangle sum to 180°. Therefore:
[tex]B = 180^{\circ}-A-C\\\\B = 180^{\circ}-24.3^{\circ}-54.6^{\circ}\\\\B = 101.1^{\circ}[/tex]
Substitute the values of A, C, and c into the Law of Sines equation and solve for side length a:
[tex]\dfrac{a}{\sin 24.3^{\circ}}=\dfrac{2.68}{\sin 54.6^{\circ}}\\\\\\a=\dfrac{2.68\sin 24.3^{\circ}}{\sin 54.6^{\circ}}\\\\\\a=1.352988435...\\\\\\a=1.35\;\sf (2\;d.p.)[/tex]
Substitute the values of B, C and c into the Law of Sines equation and solve for side length b:
[tex]\dfrac{b}{\sin 101.1^{\circ}}=\dfrac{2.68}{\sin 54.6^{\circ}}\\\\\\b=\dfrac{2.68\sin 101.1^{\circ}}{\sin 54.6^{\circ}}\\\\\\b=3.226321508...\\\\\\b=3.23\; \sf (2\;d.p.)[/tex]
