Answer:
16. Angle C is approximately 13.0 degrees.
17. The length of segment BC is approximately 45.0.
18. Angle B is approximately 26.0 degrees.
15. The length of segment DF "e" is approximately 12.9.
Step-by-step explanation:
16
By the law of sine, the sine of interior angles of a triangle are proportional to the length of the side opposite to that angle.
For triangle ABC:
- [tex]\sin{A} = \sin{103\textdegree{}}[/tex],
- The opposite side of angle A [tex]a = BC = 26[/tex],
- The angle C is to be found, and
- The length of the side opposite to angle C [tex]c = AB = 6[/tex].
[tex]\displaystyle \frac{\sin{C}}{\sin{A}} = \frac{c}{a}[/tex].
[tex]\displaystyle \sin{C} = \frac{c}{a}\cdot \sin{A} = \frac{6}{26}\times \sin{103\textdegree}[/tex].
[tex]\displaystyle C = \sin^{-1}{(\sin{C}}) = \sin^{-1}{\left(\frac{c}{a}\cdot \sin{A}\right)} = \sin^{-1}{\left(\frac{6}{26}\times \sin{103\textdegree}}\right)} = 13.0\textdegree{}[/tex].
Note that the inverse sine function here [tex]\sin^{-1}()[/tex] is also known as arcsin.
17
By the law of cosine,
[tex]c^{2} = a^{2} + b^{2} - 2\;a\cdot b\cdot \cos{C}[/tex],
where
- [tex]a[/tex], [tex]b[/tex], and [tex]c[/tex] are the lengths of sides of triangle ABC, and
- [tex]\cos{C}[/tex] is the cosine of angle C.
For triangle ABC:
- [tex]b = 21[/tex],
- [tex]c = 30[/tex],
- The length of [tex]a[/tex] (segment BC) is to be found, and
- The cosine of angle A is [tex]\cos{123\textdegree}[/tex].
Therefore, replace C in the equation with A, and the law of cosine will become:
[tex]a^{2} = b^{2} + c^{2} - 2\;b\cdot c\cdot \cos{A}[/tex].
[tex]\displaystyle \begin{aligned}a &= \sqrt{b^{2} + c^{2} - 2\;b\cdot c\cdot \cos{A}}\\&=\sqrt{21^{2} + 30^{2} - 2\times 21\times 30 \times \cos{123\textdegree}}\\&=45.0 \end{aligned}[/tex].
18
For triangle ABC:
- [tex]a = 14[/tex],
- [tex]b = 9[/tex],
- [tex]c = 6[/tex], and
- Angle B is to be found.
Start by finding the cosine of angle B. Apply the law of cosine.
[tex]b^{2} = a^{2} + c^{2} - 2\;a\cdot c\cdot \cos{B}[/tex].
[tex]\displaystyle \cos{B} = \frac{a^{2} + c^{2} - b^{2}}{2\;a\cdot c}[/tex].
[tex]\displaystyle B = \cos^{-1}{\left(\frac{a^{2} + c^{2} - b^{2}}{2\;a\cdot c}\right)} = \cos^{-1}{\left(\frac{14^{2} + 6^{2} - 9^{2}}{2\times 14\times 6}\right)} = 26.0\textdegree[/tex].
15
For triangle DEF:
- The length of segment DF is to be found,
- The length of segment EF is 9,
- The sine of angle E is [tex]\sin{64\textdegree}}[/tex], and
- The sine of angle D is [tex]\sin{39\textdegree}[/tex].
Apply the law of sine:
[tex]\displaystyle \frac{DF}{EF} = \frac{\sin{E}}{\sin{D}}[/tex]
[tex]\displaystyle DF = \frac{\sin{E}}{\sin{D}}\cdot EF = \frac{\sin{64\textdegree}}{39\textdegree} \times 9 = 12.9[/tex].
