# Oppgaver: Sinussetningen :::::::::::::::{exercise} Oppgave 1 --- level: 1 --- En trekant $\triangle ABC$ er vist i figuren nedenfor. :::{figure} ./figurer/oppgaver/oppgave_1/figur.svg --- width: 80% class: no-click, adaptive-figure --- ::: :::{cas-popup} 350 500 ::: ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Bestem $a$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ a \approx 1.39 $$ :::: ::::{admonition} Løsning --- class: solution, dropdown --- :::{figure} ./figurer/oppgaver/oppgave_1/a/sol.png --- width: 70% class: no-click, adaptive-figure --- ::: :::: ::::::::::::: :::::::::::::{tab-item} b Bestem $c$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ c = 2 $$ :::: ::::{admonition} Løsning --- class: solution, dropdown --- :::{figure} ./figurer/oppgaver/oppgave_1/b/sol.png --- width: 70% class: no-click, adaptive-figure --- ::: :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 2 --- level: 1 --- I figuren nedenfor vises en trekant $\triangle ABC$. :::{figure} ./figurer/oppgaver/oppgave_2/figur.svg --- width: 80% class: no-click, adaptive-figure --- ::: :::{cas-popup} 350 500 ::: ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Bestem $\angle B$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ \angle B \approx 88.39\degree $$ :::: ::::{solution} :::{figure} ./figurer/oppgaver/oppgave_2/a/sol.png --- width: 80% class: no-click, adaptive-figure --- ::: Fra figuren kan vi se at $\angle B < 90 \degree$. Dermed er $$ \angle B \approx 88.39\degree. $$ :::: ::::::::::::: :::::::::::::{tab-item} b Bestem $BC$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ BC \approx 2.29 $$ :::: ::::{solution} :::{figure} ./figurer/oppgaver/oppgave_2/b/sol.png --- width: 80% class: no-click, adaptive-figure --- ::: Altså er $$ BC \approx 2.29 $$ :::: ::::::::::::: :::::::::::::{tab-item} c Bestem arealet av $\triangle ABC$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ T \approx 2.29 $$ :::: ::::{solution} :::{figure} ./figurer/oppgaver/oppgave_2/b/sol.png --- width: 80% class: no-click, adaptive-figure --- ::: som betyr at arealet av $\triangle ABC$ er $$ T = \approx 2.29 $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 3 --- level: 2 --- I $\triangle ABC$ er $\angle A = 45 \degree$, $BC = 6$ og $AC = 8$. :::{cas-popup} 350 500 ::: ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Bestem hvilke mulige vinkler $\angle B$ kan ha. ::::{admonition} Fasit --- class: answer, dropdown --- $$ \angle B \approx 70.53 \degree \or \angle B \approx 109.47 \degree $$ :::: ::::{solution} Vinkelen $\angle B$ kan enten være spiss eller stump. Vi bruker sinussetningen for å bestemme hvilke mulige verdier $\angle B$ kan ha: :::{figure} ./figurer/oppgaver/oppgave_3/a/sol.png --- width: 80% class: no-click, adaptive-figure --- ::: som betyr at $$ \angle B \approx 70.53 \degree \or \angle B \approx 109.47 \degree. $$ :::: ::::::::::::: :::::::::::::{tab-item} b Bestem hvilke to lengder $AB$ kan ha. ::::{admonition} Fasit --- class: answer, dropdown --- $$ AB \approx 3.66 \or AB \approx 7.66 $$ :::: ::::{solution} Lengden til $AB$ vil være avhengig av $\angle B$. La oss først anta $\angle B \approx 70.53 \degree$. Da kan vi bruke sinussetningen til å bestemme den ene lengden $AB$ kan ha: :::{figure} ./figurer/oppgaver/oppgave_3/b/sol_1.png --- width: 70% class: no-click, adaptive-figure --- ::: Altså kan vi ha $$ \angle B \approx 70.53 \degree \and AB \approx 7.66. $$ Den andre mulige verdien for $\angle B$ er $\angle B \approx 109.47 \degree$. Da kan vi bruke sinussetningen til å bestemme den andre lengden $AB$ kan ha: :::{figure} ./figurer/oppgaver/oppgave_3/b/sol_2.png --- width: 70% class: no-click, adaptive-figure --- ::: Altså kan vi ha $$ \angle B \approx 109.47 \degree \and AB \approx 3.66. $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 4 --- level: 2 --- I figuren nedenfor vises $\square ABCD$. :::{figure} ./figurer/oppgaver/oppgave_4/figur.svg --- width: 80% class: no-click, adaptive-figure --- ::: :::{cas-popup} 350 500 ::: ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Bestem $\angle BDA$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ \angle BDA \approx 26.57\degree. $$ :::: ::::{solution} Vi bruker sinussetningen til å bestemme $\angle BDA$: :::{figure} ./figurer/oppgaver/oppgave_4/a/sol.png --- width: 70% class: no-click, adaptive-figure --- ::: Altså er $$ \angle BDA \approx 26.57\degree. $$ :::: ::::::::::::: :::::::::::::{tab-item} b Bestem arealet $T$ av $\square ABCD$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ T_{ABCD} \approx 22.5 $$ :::: ::::{solution} Vi bruker arealsetningen på de trekantene $\triangle ABD$ og $\triangle BCD$ for å bestemme arealet $T$ av $\square ABCD$: :::{figure} ./figurer/oppgaver/oppgave_4/b/sol.png --- width: 70% class: no-click, adaptive-figure --- ::: Altså er $$ T_{ABCD} \approx 22.5 $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 5 --- level: 2 --- I figuren nedenfor vises $\square ABCD$. :::{figure} ./figurer/oppgaver/oppgave_5/figur.svg --- width: 80% class: no-click, adaptive-figure --- ::: :::{cas-popup} 350 500 ::: ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Bestem en eksakt verdi for $CD$ uttrykt ved $a$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ CD = \dfrac{\sqrt{3}}{2} a $$ :::: ::::{solution} Vi bruker sinussetningen til å bestemme lengden $CD$: :::{figure} ./figurer/oppgaver/oppgave_5/a/sol.png --- width: 70% class: no-click, adaptive-figure --- ::: Altså er $$ CD = \dfrac{\sqrt{3}}{2} a. $$ :::: ::::::::::::: :::::::::::::{tab-item} b Bestem en eksakt verdi for arealet $T$ av $\square ABCD$ uttrykt ved $a$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ T = \dfrac{1}{8}a^2 \left(2\sqrt{3} + 1\right) $$ :::: ::::{solution} Vi bruker arealsetningen til å bestemme arealet av de to trekantene. Vi trenger én side til i $\triangle ABD$, så vi finner $AD$ ved hjelp av sinussetningen. Vi gjør utregningen med CAS: :::{figure} ./figurer/oppgaver/oppgave_5/b/sol.png --- width: 70% --- ::: Altså er $$ T_{ABCD} = \dfrac{1}{8}a^2 \left(2\sqrt{3} + 1\right). $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 6 --- level: 2 --- I figuren nedenfor vises $\square ABCD$. :::{figure} ./figurer/oppgaver/oppgave_6/figur.svg --- width: 80% class: no-click, adaptive-figure --- ::: :::{cas-popup} 350 500 ::: ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Bestem en eksakt verdi for omkretsen $\mathcal{O}$ til $\square ABCD$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ \mathcal{O} = 8\sqrt{3} + 24 $$ :::: ::::{solution} Vi bruker først sinussetningen til å vinkelen $\angle BDA$. Vi finner at $\angle BDA = 90 \degree$ som betyr at vi kan bruke Pytagoras' setning til å bestemme $AD$. Deretter plusser vi sammen lengdene til alle sidene. :::{figure} ./figurer/oppgaver/oppgave_6/a/sol.png --- width: 70% class: no-click, adaptive-figure --- ::: Omkretsen til $\square ABCD$ er da $$ \mathcal{O} = 8\sqrt{3} + 24 $$ :::: ::::::::::::: :::::::::::::{tab-item} b Bestem en eksakt verdi for arealet $T_{ABCD}$ til $\square ABCD$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ T_{ABCD} = 32\sqrt{3} $$ :::: ::::{solution} Vi bruker arealsetningen på hver av trekantene og legger sammen arealene: :::{figure} ./figurer/oppgaver/oppgave_6/b/sol.png --- width: 70% class: no-click, adaptive-figure --- ::: Altså er arealet til $\square ABCD$: $$ T_{ABCD} = 32\sqrt{3} $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 7 --- level: 2 --- I figuren nedenfor vises $\square ABCD$. :::{figure} ./figurer/oppgaver/oppgave_7/figur.svg --- width: 80% class: no-click, adaptive-figure --- ::: :::{cas-popup} 350 500 ::: ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Bestem lengden av diagonalen $BD$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ BD = 7. $$ :::: ::::{solution} Vi bruker sinussetningen til å bestemme lengden $x = BD$: :::{figure} ./figurer/oppgaver/oppgave_7/a/sol.png --- width: 70% class: no-click, adaptive-figure --- ::: Altså er $$ BD = 7 $$ :::: ::::::::::::: :::::::::::::{tab-item} b Bestem arealet $T$ av $\square ABCD$. ::::{admonition} Fasit --- class: answer, dropdown --- $$ T \approx 29.3 $$ :::: ::::{solution} Vi bruker arealsetningen på de to trekantene og legger sammen arealene: :::{figure} ./figurer/oppgaver/oppgave_7/b/sol.png --- width: 70% class: no-click, adaptive-figure --- ::: Altså er arealet $T$ av $\square ABCD$: $$ T \approx 29.3 $$ :::: ::::::::::::: :::::::::::::: :::::::::::::::