Trigonométrie - STI2D/STL

Le cercle trigonométrique

Exercice 1 : 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 \(\dfrac{2\pi }{3}\).

{"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": "-.."}]]}

Exercice 2 : Convertir des degrés en radians (valeur exacte)

Convertir l'angle suivant en radians. \[146^{\circ}\] Il ne faut pas repréciser l'unité. (On précisera la réponse avec la valeur \(\pi\) disponible en cliquant sur le clavier bleu.)

Exercice 3 : Retrouver le point correspondant à un angle dans le cercle trignometrique (mesure entre -pi 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", {}], [[0.8660254037844386, -0.5], "I", "below right", {}], [[0.7071067811865476, -0.7071067811865476], "J", "below right", {}], [[0.5, -0.8660254037844386], "K", "below right", {}], [[0.0, -1.0], "L", "below left", {}], [[-0.5, -0.8660254037844386], "M", "below left", {}], [[-0.7071067811865476, -0.7071067811865476], "N", "below left", {}], [[-0.8660254037844386, -0.5], "O", "below left", {}], [[-1.0, 0.0], "P", "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"}], [[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": "-.."}]]}

Exercice 4 : 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": [[[0.5, 0.8660254037844386], "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": [[[0.5, 0.8660254037844386], 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 5 : Retrouver les réels correspondants à la même image dans dans le cercle trigonométrique

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

\(A = \frac{19\pi }{2}\)
\(B = \frac{41\pi }{4}\)
\(C = \frac{3\pi }{4}\)
\(D = \frac{21\pi }{4}\)
\(E = \frac{3\pi }{2}\)

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

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