Fonctions cosinus et sinus - Spécialité

Révisions : Angles remarquables et cercle trigonométrique

Exercice 1 : Retrouver l'angle correspondant a un point correspondantle cercle trignometrique (mesure entre 0 et pi)

Sur le cercle trigonométrique suivant, Donner le réel dont le point A est l'image.

{"init": {"range": [[-1.2, 1.2], [-1.2, 1.2]], "scale": [250, 250]}, "label": [[[0.8660254037844386, 0.5], "A", "above right", {}], [[0.7071067811865476, 0.0], "\\dfrac{\\sqrt{2}}{2}", "above left", {}], [[0.8660254037844386, 0.0], "\\dfrac{\\sqrt{3}}{2}", "above left", {}], [[0.5, 0.0], "\\dfrac{1}{2}", "above left", {}], [[-0.7071067811865476, 0.0], "- \\dfrac{\\sqrt{2}}{2}", "above right", {}], [[-0.8660254037844386, 0.0], "- \\dfrac{\\sqrt{3}}{2}", "above right", {}], [[-0.5, 0.0], "- \\dfrac{1}{2}", "above right", {}], [[0.0, 0.7071067811865476], "\\dfrac{\\sqrt{2}}{2}", "below right", {}], [[0.0, 0.5], "\\dfrac{1}{2}", "below right", {}], [[0.0, 0.8660254037844386], "\\dfrac{\\sqrt{3}}{2}", "below right", {}], [[0.0, -0.7071067811865476], "- \\dfrac{\\sqrt{2}}{2}", "above right", {}], [[0.0, -0.5], "- \\dfrac{1}{2}", "above right", {}], [[0.0, -0.8660254037844386], "- \\dfrac{\\sqrt{3}}{2}", "above right", {}]], "circle": [[[0.8660254037844386, 0.5], 0.01, {"fill": "black"}], [[0.0, 0.0], 1.0, {}], [[0.7071067811865476, 0.0], 0.01, {"fill": "black"}], [[-0.7071067811865476, 0.0], 0.01, {"fill": "black"}], [[0.0, 0.7071067811865476], 0.01, {"fill": "black"}], [[0.0, -0.7071067811865476], 0.01, {"fill": "black"}], [[0.8660254037844386, 0.0], 0.01, {"fill": "black"}], [[-0.8660254037844386, 0.0], 0.01, {"fill": "black"}], [[0.0, 0.5], 0.01, {"fill": "black"}], [[0.0, -0.5], 0.01, {"fill": "black"}], [[0.5, 0.0], 0.01, {"fill": "black"}], [[-0.5, 0.0], 0.01, {"fill": "black"}], [[0.0, 0.8660254037844386], 0.01, {"fill": "black"}], [[0.0, -0.8660254037844386], 0.01, {"fill": "black"}]], "line": [[[-1.2, 0.0], [1.2, 0.0], {"subtype": "segment", "arrow-end": "classic-wide-long"}], [[0.0, -1.2], [0.0, 1.2], {"subtype": "segment", "arrow-end": "classic-wide-long"}], [[0.7071067811865476, 0.7071067811865476], [0.7071067811865476, -0.7071067811865476], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.7071067811865476, 0.7071067811865476], [-0.7071067811865476, 0.7071067811865476], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.7071067811865476, -0.7071067811865476], [-0.7071067811865476, -0.7071067811865476], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[-0.7071067811865476, 0.7071067811865476], [-0.7071067811865476, -0.7071067811865476], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.8660254037844386, 0.5], [0.8660254037844386, -0.5], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.8660254037844386, 0.5], [-0.8660254037844386, 0.5], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.8660254037844386, -0.5], [-0.8660254037844386, -0.5], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[-0.8660254037844386, 0.5], [-0.8660254037844386, -0.5], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.5, 0.8660254037844386], [0.5, -0.8660254037844386], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.5, 0.8660254037844386], [-0.5, 0.8660254037844386], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.5, -0.8660254037844386], [-0.5, -0.8660254037844386], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[-0.5, 0.8660254037844386], [-0.5, -0.8660254037844386], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}]]}

(On donnera une valeur dans l'intervalle \(\left[0; \pi\right]\)

Exercice 2 : Retrouver l'angle correspondant a un point correspondantle cercle trignometrique (mesure entre 0 et pi/2)

Sur le cercle trigonométrique suivant, Donner le réel dont le point A est l'image.

{"init": {"range": [[-1.2, 1.2], [-1.2, 1.2]], "scale": [250, 250]}, "label": [[[0.0, 1.0], "A", "above", {}], [[0.7071067811865476, 0.0], "\\dfrac{\\sqrt{2}}{2}", "above left", {}], [[0.8660254037844386, 0.0], "\\dfrac{\\sqrt{3}}{2}", "above left", {}], [[0.5, 0.0], "\\dfrac{1}{2}", "above left", {}], [[-0.7071067811865476, 0.0], "- \\dfrac{\\sqrt{2}}{2}", "above right", {}], [[-0.8660254037844386, 0.0], "- \\dfrac{\\sqrt{3}}{2}", "above right", {}], [[-0.5, 0.0], "- \\dfrac{1}{2}", "above right", {}], [[0.0, 0.7071067811865476], "\\dfrac{\\sqrt{2}}{2}", "below right", {}], [[0.0, 0.5], "\\dfrac{1}{2}", "below right", {}], [[0.0, 0.8660254037844386], "\\dfrac{\\sqrt{3}}{2}", "below right", {}], [[0.0, -0.7071067811865476], "- \\dfrac{\\sqrt{2}}{2}", "above right", {}], [[0.0, -0.5], "- \\dfrac{1}{2}", "above right", {}], [[0.0, -0.8660254037844386], "- \\dfrac{\\sqrt{3}}{2}", "above right", {}]], "circle": [[[0.0, 1.0], 0.01, {"fill": "black"}], [[0.0, 0.0], 1.0, {}], [[0.7071067811865476, 0.0], 0.01, {"fill": "black"}], [[-0.7071067811865476, 0.0], 0.01, {"fill": "black"}], [[0.0, 0.7071067811865476], 0.01, {"fill": "black"}], [[0.0, -0.7071067811865476], 0.01, {"fill": "black"}], [[0.8660254037844386, 0.0], 0.01, {"fill": "black"}], [[-0.8660254037844386, 0.0], 0.01, {"fill": "black"}], [[0.0, 0.5], 0.01, {"fill": "black"}], [[0.0, -0.5], 0.01, {"fill": "black"}], [[0.5, 0.0], 0.01, {"fill": "black"}], [[-0.5, 0.0], 0.01, {"fill": "black"}], [[0.0, 0.8660254037844386], 0.01, {"fill": "black"}], [[0.0, -0.8660254037844386], 0.01, {"fill": "black"}]], "line": [[[-1.2, 0.0], [1.2, 0.0], {"subtype": "segment", "arrow-end": "classic-wide-long"}], [[0.0, -1.2], [0.0, 1.2], {"subtype": "segment", "arrow-end": "classic-wide-long"}], [[0.7071067811865476, 0.7071067811865476], [0.7071067811865476, -0.7071067811865476], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.7071067811865476, 0.7071067811865476], [-0.7071067811865476, 0.7071067811865476], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.7071067811865476, -0.7071067811865476], [-0.7071067811865476, -0.7071067811865476], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[-0.7071067811865476, 0.7071067811865476], [-0.7071067811865476, -0.7071067811865476], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.8660254037844386, 0.5], [0.8660254037844386, -0.5], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.8660254037844386, 0.5], [-0.8660254037844386, 0.5], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.8660254037844386, -0.5], [-0.8660254037844386, -0.5], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[-0.8660254037844386, 0.5], [-0.8660254037844386, -0.5], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.5, 0.8660254037844386], [0.5, -0.8660254037844386], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.5, 0.8660254037844386], [-0.5, 0.8660254037844386], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.5, -0.8660254037844386], [-0.5, -0.8660254037844386], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[-0.5, 0.8660254037844386], [-0.5, -0.8660254037844386], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}]]}

On donnera une valeur dans l'intervalle \(\left[0; \frac{\pi}{2}\right]\)

Exercice 3 : Retrouver le point correspondant à un angle dans le cercle trignometrique (mesure entre 0 et pi/2)

Sur le cercle trigonométrique suivant, Donner le nom du point image du nombre \(\dfrac{\pi }{4}\).

{"init": {"range": [[-1.2, 1.2], [-1.2, 1.2]], "scale": [250, 250]}, "label": [[[1.0, 0.0], "A", "above right", {}], [[0.8660254037844386, 0.5], "B", "above right", {}], [[0.7071067811865476, 0.7071067811865476], "C", "above right", {}], [[0.5, 0.8660254037844386], "D", "above right", {}], [[0.0, 1.0], "E", "above left", {}], [[0.7071067811865476, 0.0], "\\dfrac{\\sqrt{2}}{2}", "above left", {}], [[0.8660254037844386, 0.0], "\\dfrac{\\sqrt{3}}{2}", "above left", {}], [[0.5, 0.0], "\\dfrac{1}{2}", "above left", {}], [[-0.7071067811865476, 0.0], "- \\dfrac{\\sqrt{2}}{2}", "above right", {}], [[-0.8660254037844386, 0.0], "- \\dfrac{\\sqrt{3}}{2}", "above right", {}], [[-0.5, 0.0], "- \\dfrac{1}{2}", "above right", {}], [[0.0, 0.7071067811865476], "\\dfrac{\\sqrt{2}}{2}", "below right", {}], [[0.0, 0.5], "\\dfrac{1}{2}", "below right", {}], [[0.0, 0.8660254037844386], "\\dfrac{\\sqrt{3}}{2}", "below right", {}], [[0.0, -0.7071067811865476], "- \\dfrac{\\sqrt{2}}{2}", "above right", {}], [[0.0, -0.5], "- \\dfrac{1}{2}", "above right", {}], [[0.0, -0.8660254037844386], "- \\dfrac{\\sqrt{3}}{2}", "above right", {}]], "circle": [[[1.0, 0.0], 0.01, {"fill": "black"}], [[0.8660254037844386, 0.5], 0.01, {"fill": "black"}], [[0.7071067811865476, 0.7071067811865476], 0.01, {"fill": "black"}], [[0.5, 0.8660254037844386], 0.01, {"fill": "black"}], [[0.0, 1.0], 0.01, {"fill": "black"}], [[0.0, 0.0], 1.0, {}], [[0.7071067811865476, 0.0], 0.01, {"fill": "black"}], [[-0.7071067811865476, 0.0], 0.01, {"fill": "black"}], [[0.0, 0.7071067811865476], 0.01, {"fill": "black"}], [[0.0, -0.7071067811865476], 0.01, {"fill": "black"}], [[0.8660254037844386, 0.0], 0.01, {"fill": "black"}], [[-0.8660254037844386, 0.0], 0.01, {"fill": "black"}], [[0.0, 0.5], 0.01, {"fill": "black"}], [[0.0, -0.5], 0.01, {"fill": "black"}], [[0.5, 0.0], 0.01, {"fill": "black"}], [[-0.5, 0.0], 0.01, {"fill": "black"}], [[0.0, 0.8660254037844386], 0.01, {"fill": "black"}], [[0.0, -0.8660254037844386], 0.01, {"fill": "black"}]], "line": [[[-1.2, 0.0], [1.2, 0.0], {"subtype": "segment", "arrow-end": "classic-wide-long"}], [[0.0, -1.2], [0.0, 1.2], {"subtype": "segment", "arrow-end": "classic-wide-long"}], [[0.7071067811865476, 0.7071067811865476], [0.7071067811865476, -0.7071067811865476], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.7071067811865476, 0.7071067811865476], [-0.7071067811865476, 0.7071067811865476], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.7071067811865476, -0.7071067811865476], [-0.7071067811865476, -0.7071067811865476], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[-0.7071067811865476, 0.7071067811865476], [-0.7071067811865476, -0.7071067811865476], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.8660254037844386, 0.5], [0.8660254037844386, -0.5], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.8660254037844386, 0.5], [-0.8660254037844386, 0.5], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.8660254037844386, -0.5], [-0.8660254037844386, -0.5], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[-0.8660254037844386, 0.5], [-0.8660254037844386, -0.5], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.5, 0.8660254037844386], [0.5, -0.8660254037844386], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.5, 0.8660254037844386], [-0.5, 0.8660254037844386], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.5, -0.8660254037844386], [-0.5, -0.8660254037844386], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[-0.5, 0.8660254037844386], [-0.5, -0.8660254037844386], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}]]}

Exercice 4 : Retrouver les réels correspondants à la même image dans dans le cercle trigonométrique

Soit les réels A, B, C, D, E tels que :

\(A = \frac{44\pi }{3}\)
\(B = \frac{22\pi }{3}\)
\(C = 3\pi \)
\(D = \frac{20\pi }{3}\)
\(E = \frac{37\pi }{3}\)

Parmi ces réels, lesquels ont le même point image sur le cercle trigonométrique ?

Exercice 5 : Retrouver le point correspondant à un angle dans le cercle trignometrique (mesure entre 0 et pi)

Sur le cercle trigonométrique suivant, Donner le nom du point image du nombre \(\dfrac{\pi }{6}\).

{"init": {"range": [[-1.2, 1.2], [-1.2, 1.2]], "scale": [250, 250]}, "label": [[[1.0, 0.0], "A", "above right", {}], [[0.8660254037844386, 0.5], "B", "above right", {}], [[0.7071067811865476, 0.7071067811865476], "C", "above right", {}], [[0.5, 0.8660254037844386], "D", "above right", {}], [[0.0, 1.0], "E", "above left", {}], [[-0.5, 0.8660254037844386], "F", "above left", {}], [[-0.7071067811865476, 0.7071067811865476], "G", "above left", {}], [[-0.8660254037844386, 0.5], "H", "above left", {}], [[-1.0, 0.0], "I", "above left", {}], [[0.7071067811865476, 0.0], "\\dfrac{\\sqrt{2}}{2}", "above left", {}], [[0.8660254037844386, 0.0], "\\dfrac{\\sqrt{3}}{2}", "above left", {}], [[0.5, 0.0], "\\dfrac{1}{2}", "above left", {}], [[-0.7071067811865476, 0.0], "- \\dfrac{\\sqrt{2}}{2}", "above right", {}], [[-0.8660254037844386, 0.0], "- \\dfrac{\\sqrt{3}}{2}", "above right", {}], [[-0.5, 0.0], "- \\dfrac{1}{2}", "above right", {}], [[0.0, 0.7071067811865476], "\\dfrac{\\sqrt{2}}{2}", "below right", {}], [[0.0, 0.5], "\\dfrac{1}{2}", "below right", {}], [[0.0, 0.8660254037844386], "\\dfrac{\\sqrt{3}}{2}", "below right", {}], [[0.0, -0.7071067811865476], "- \\dfrac{\\sqrt{2}}{2}", "above right", {}], [[0.0, -0.5], "- \\dfrac{1}{2}", "above right", {}], [[0.0, -0.8660254037844386], "- \\dfrac{\\sqrt{3}}{2}", "above right", {}]], "circle": [[[1.0, 0.0], 0.01, {"fill": "black"}], [[0.8660254037844386, 0.5], 0.01, {"fill": "black"}], [[0.7071067811865476, 0.7071067811865476], 0.01, {"fill": "black"}], [[0.5, 0.8660254037844386], 0.01, {"fill": "black"}], [[0.0, 1.0], 0.01, {"fill": "black"}], [[-0.5, 0.8660254037844386], 0.01, {"fill": "black"}], [[-0.7071067811865476, 0.7071067811865476], 0.01, {"fill": "black"}], [[-0.8660254037844386, 0.5], 0.01, {"fill": "black"}], [[-1.0, 0.0], 0.01, {"fill": "black"}], [[0.0, 0.0], 1.0, {}], [[0.7071067811865476, 0.0], 0.01, {"fill": "black"}], [[-0.7071067811865476, 0.0], 0.01, {"fill": "black"}], [[0.0, 0.7071067811865476], 0.01, {"fill": "black"}], [[0.0, -0.7071067811865476], 0.01, {"fill": "black"}], [[0.8660254037844386, 0.0], 0.01, {"fill": "black"}], [[-0.8660254037844386, 0.0], 0.01, {"fill": "black"}], [[0.0, 0.5], 0.01, {"fill": "black"}], [[0.0, -0.5], 0.01, {"fill": "black"}], [[0.5, 0.0], 0.01, {"fill": "black"}], [[-0.5, 0.0], 0.01, {"fill": "black"}], [[0.0, 0.8660254037844386], 0.01, {"fill": "black"}], [[0.0, -0.8660254037844386], 0.01, {"fill": "black"}]], "line": [[[-1.2, 0.0], [1.2, 0.0], {"subtype": "segment", "arrow-end": "classic-wide-long"}], [[0.0, -1.2], [0.0, 1.2], {"subtype": "segment", "arrow-end": "classic-wide-long"}], [[0.7071067811865476, 0.7071067811865476], [0.7071067811865476, -0.7071067811865476], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.7071067811865476, 0.7071067811865476], [-0.7071067811865476, 0.7071067811865476], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.7071067811865476, -0.7071067811865476], [-0.7071067811865476, -0.7071067811865476], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[-0.7071067811865476, 0.7071067811865476], [-0.7071067811865476, -0.7071067811865476], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.8660254037844386, 0.5], [0.8660254037844386, -0.5], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.8660254037844386, 0.5], [-0.8660254037844386, 0.5], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.8660254037844386, -0.5], [-0.8660254037844386, -0.5], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[-0.8660254037844386, 0.5], [-0.8660254037844386, -0.5], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.5, 0.8660254037844386], [0.5, -0.8660254037844386], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.5, 0.8660254037844386], [-0.5, 0.8660254037844386], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[0.5, -0.8660254037844386], [-0.5, -0.8660254037844386], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}], [[-0.5, 0.8660254037844386], [-0.5, -0.8660254037844386], {"subtype": "segment", "stroke": "#6495ed", "stroke-dasharray": "-.."}]]}

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