# Oppgaver: Enhetssirkelen :::::::::::::::{exercise} Oppgave 1 ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a :::{plot} align: right width: 100% circle: (0, 0), 1, blue, solid let: v = 36 * pi / 180 line-segment: (0, 0), (cos(v), sin(v)), red point: (cos(v), sin(v)) axis: equal text: cos(v), sin(v), "({cos(v):.2f}, {sin(v):.2f})", top-right angle-arc: (0, 0), 0.3, 0, 36, red text: 0.35 * cos(v / 2), 0.25 * sin(v / 2), "$36^\circ$", top-right fontsize: 28 grid: off ::: Bruk figuren til høyre til å bestemme * $\cos 36\degree$ * $\sin 36\degree$ * $\tan 36\degree$ :::{clear} ::: ::::{answer} * $\cos 36 \degree = 0.81$ * $\sin 36 \degree = 0.59$ * $\tan 36 \degree = \dfrac{0.59}{0.81} = \dfrac{59}{81}$ :::: ::::::::::::: :::::::::::::{tab-item} b :::{plot} align: right width: 100% circle: (0, 0), 1, blue, solid let: v = 72 * pi / 180 line-segment: (0, 0), (cos(v), sin(v)), red point: (cos(v), sin(v)) axis: equal text: cos(v), sin(v), "({cos(v):.2f}, {sin(v):.2f})", top-right angle-arc: (0, 0), 0.3, 0, 72, red text: 0.35 * cos(v / 2), 0.25 * sin(v / 2), "$72^\circ$", top-right fontsize: 28 grid: off ::: Bruk figuren til å bestemme * $\cos 72\degree$ * $\sin 72\degree$ * $\tan 72\degree$ :::{clear} ::: ::::{answer} * $\cos 72\degree = 0.31$ * $\sin 72\degree = 0.95$ * $\tan 72\degree = \dfrac{0.95}{0.31} = \dfrac{95}{31}$ :::: ::::::::::::: :::::::::::::{tab-item} c :::{plot} align: right width: 100% circle: (0, 0), 1, blue, solid let: v = 140 * pi / 180 line-segment: (0, 0), (cos(v), sin(v)), red point: (cos(v), sin(v)) axis: equal text: cos(v), sin(v), "({cos(v):.2f}, {sin(v):.2f})", top-left angle-arc: (0, 0), 0.3, 0, 140, red text: 0.35 * cos(v / 2), 0.25 * sin(v / 2), "$140^\circ$", top-right fontsize: 28 grid: off ::: Bruk figuren til å bestemme * $\cos 140\degree$ * $\sin 140\degree$ * $\tan 140\degree$ :::{clear} ::: ::::{answer} * $\cos 140\degree = -0.77$ * $\sin 140\degree = 0.64$ * $\tan 140\degree = \dfrac{0.64}{-0.77} = -\dfrac{64}{77}$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 2 ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a :::{plot} align: right width: 100% circle: (0, 0), 1, blue, solid let: v = 30 * pi / 180 line-segment: (0, 0), (cos(v), sin(v)), red point: (cos(v), sin(v)) axis: equal text: cos(v), sin(v), "$P(x, y)$", top-right angle-arc: (0, 0), 0.3, 0, 30, red text: 0.35 * cos(v / 2), 0.2 * sin(v / 2), "$30^\circ$", top-right fontsize: 28 grid: off ::: Bestem koordinatene til punktet $P$ på enhetssirkelen vist til høyre. :::{clear} ::: ::::{answer} $$ P\left(\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right) $$ :::: ::::::::::::: :::::::::::::{tab-item} b :::{plot} align: right width: 100% circle: (0, 0), 1, blue, solid let: v = 45 * pi / 180 line-segment: (0, 0), (cos(v), sin(v)), red point: (cos(v), sin(v)) axis: equal text: cos(v), sin(v), "$P(x, y)$", top-right angle-arc: (0, 0), 0.3, 0, 45, red text: 0.35 * cos(v / 2), 0.2 * sin(v / 2), "$45^\circ$", top-right fontsize: 28 grid: off ::: Bestem koordinatene til punktet $P$ på enhetssirkelen vist til høyre. :::{clear} ::: ::::{answer} $$ P\left(\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right) $$ :::: ::::::::::::: :::::::::::::{tab-item} c :::{plot} align: right width: 100% circle: (0, 0), 1, blue, solid let: v = 60 * pi / 180 line-segment: (0, 0), (cos(v), sin(v)), red point: (cos(v), sin(v)) axis: equal text: cos(v), sin(v), "$P(x, y)$", top-right angle-arc: (0, 0), 0.3, 0, 60, red text: 0.35 * cos(v / 2), 0.2 * sin(v / 2), "$60^\circ$", top-right fontsize: 28 grid: off ::: Bestem koordinatene til punktet $P$ på enhetssirkelen vist til høyre. :::{clear} ::: ::::{answer} $$ P\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right) $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 3 :::{plot} width: 60% circle: (0, 0), 1, blue, solid let: v = 120 * pi / 180 line-segment: (0, 0), (cos(v), sin(v)), red let: ds = 0.15 line-segment: (cos(v), sin(v)), (0, sin(v)), red, dashed line-segment: (0, sin(v) - ds), (-ds, sin(v) - ds), solid, gray line-segment: (-ds, sin(v) - ds), (-ds, sin(v)), solid, gray point: (cos(v), sin(v)) axis: equal text: cos(v), sin(v), "$P(x, y)$", top-left angle-arc: (0, 0), 0.3, 0, 120, red text: 0.35 * cos(v / 2), 0.2 * sin(v / 2), "$120^\circ$", top-right fontsize: 28 grid: off nocache: ticks: off ::: I enhetssirkelen ovenfor er det tegnet inn et punkt $P$ som ligger på sirkelen med en vinkel på $120\degree$ med $x$-aksen. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Bestem koordinatene til punktet $P$ på enhetssirkelen vist ovenfor. ::::{answer} $$ P\left(-\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right) $$ :::: ::::::::::::: :::::::::::::{tab-item} b Bestem eksakte verdier for * $\cos 120\degree$ * $\sin 120\degree$ * $\tan 120\degree$ ::::{answer} $$ \cos 120\degree = -\dfrac{1}{2} $$ $$ \sin 120 \degree = \dfrac{\sqrt{3}}{2} $$ $$ \tan 120 \degree = -\sqrt{3} $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 4 :::{plot} width: 60% circle: (0, 0), 1, blue, solid let: v = 135 * pi / 180 line-segment: (0, 0), (cos(v), sin(v)), red let: ds = 0.15 line-segment: (cos(v), sin(v)), (0, sin(v)), red, dashed line-segment: (0, sin(v) - ds), (-ds, sin(v) - ds), solid, gray line-segment: (-ds, sin(v) - ds), (-ds, sin(v)), solid, gray point: (cos(v), sin(v)) axis: equal text: cos(v), sin(v), "$P(x, y)$", top-left angle-arc: (0, 0), 0.3, 0, 135, red text: 0.35 * cos(v / 2), 0.25 * sin(v / 2), "$135^\circ$", top-right fontsize: 28 grid: off nocache: ticks: off ::: I enhetssirkelen ovenfor er det tegnet inn et punkt $P$ som ligger på sirkelen med en vinkel på $135\degree$ med $x$-aksen. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Bestem koordinatene til punktet $P$. ::::{answer} $$ P\left(-\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right) $$ :::: ::::::::::::: :::::::::::::{tab-item} b Bestem eksakte verdier for * $\cos 135\degree$ * $\sin 135\degree$ * $\tan 135\degree$ ::::{answer} * $\cos 135\degree = -\dfrac{\sqrt{2}}{2}$ * $\sin 135\degree = \dfrac{\sqrt{2}}{2}$ * $\tan 135\degree = -1$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 5 :::{plot} width: 60% circle: (0, 0), 1, blue, solid let: v = 150 * pi / 180 line-segment: (0, 0), (cos(v), sin(v)), red let: ds = 0.15 line-segment: (cos(v), sin(v)), (0, sin(v)), red, dashed line-segment: (0, sin(v) - ds), (-ds, sin(v) - ds), solid, gray line-segment: (-ds, sin(v) - ds), (-ds, sin(v)), solid, gray point: (cos(v), sin(v)) axis: equal text: cos(v), sin(v), "$P(x, y)$", top-left angle-arc: (0, 0), 0.2, 0, 150, red text: 0.25 * cos(v / 2), 0.15 * sin(v / 2), "$150^\circ$", top-right fontsize: 28 grid: off nocache: ticks: off ::: I enhetssirkelen ovenfor er det tegnet inn et punkt $P$ som ligger på sirkelen med en vinkel på $150\degree$ med $x$-aksen. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Bestem koordinatene til punktet $P$. ::::{answer} $$ P\left(-\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right) $$ :::: ::::::::::::: :::::::::::::{tab-item} b Bestem eksakte verdier for * $\cos 150\degree$ * $\sin 150\degree$ * $\tan 150\degree$ ::::{answer} * $\cos 150\degree = -\dfrac{\sqrt{3}}{2}$ * $\sin 150\degree = \dfrac{1}{2}$ * $\tan 150\degree = -\dfrac{1}{\sqrt{3}} = -\dfrac{\sqrt{3}}{3}$ :::: ::::::::::::: :::::::::::::: :::::::::::::::