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Sagot :
Given the information that [tex]\(\cot \theta = -6\)[/tex] and [tex]\(\sec \theta < 0\)[/tex] with [tex]\(0 \leq \theta < 2\pi\)[/tex], we can find the exact values of the required trigonometric functions step-by-step.
Step 1: Find [tex]\(\tan \theta\)[/tex]
Since [tex]\(\cot \theta = -6\)[/tex], we know that:
[tex]\[ \cot \theta = \frac{1}{\tan \theta} \implies \tan \theta = \frac{1}{\cot \theta} = \frac{1}{-6} = -\frac{1}{6} \][/tex]
Therefore,
[tex]\[ \tan \theta = -0.16666666666666666 \][/tex]
Step 2: Find [tex]\(\sec \theta\)[/tex] and [tex]\(\cos \theta\)[/tex]
Given [tex]\(\sec \theta < 0\)[/tex], [tex]\(\theta\)[/tex] must be in the range where cosine is negative. This condition places [tex]\(\theta\)[/tex] in the second or third quadrant. Since we also need to satisfy the Pythagorean identity:
[tex]\[ \tan^2 \theta + 1 = \sec^2 \theta \][/tex]
we can calculate:
[tex]\[ (-0.16666666666666666)^2 + 1 = \sec^2 \theta \implies \sec^2 \theta = \frac{1}{36} + 1 = \frac{1}{36} + \frac{36}{36} = \frac{37}{36} = 1.027777777777778 \][/tex]
[tex]\[ \sec \theta = -\sqrt{1.027777777777778} = -1.0137937550497031 \][/tex]
Since [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex],
[tex]\[ \cos \theta = -1.0137937550497031^{-1} = -0.9863939238321439 \][/tex]
Step 3: Find [tex]\(\sin \theta\)[/tex]
Using the identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \implies \sin^2 \theta = 1 - \cos^2 \theta \][/tex]
[tex]\[ \sin^2 \theta = 1 - (-0.9863939238321439)^2 = 1 - 0.9730141283301396 = 0.026985871669860395 \][/tex]
[tex]\[ \sin \theta = \sqrt{0.026985871669860395} = 0.16439898730535646 \][/tex]
Step 4: Calculate [tex]\(\sin(2\theta)\)[/tex]
Using the double-angle formula for sine:
[tex]\[ \sin(2\theta) = 2 \sin \theta \cos \theta \][/tex]
[tex]\[ \sin(2\theta) = 2 \times 0.16439898730535646 \times -0.9863939238321439 = -0.32432432432432273 \][/tex]
Step 5: Calculate [tex]\(\cos(2\theta)\)[/tex]
Using the double-angle formula for cosine:
[tex]\[ \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \][/tex]
[tex]\[ \cos(2\theta) = (-0.9863939238321439)^2 - (0.16439898730535646)^2 = 0.9730141283301396 - 0.027068136054193123 = 0.9459459459459465 \][/tex]
Step 6: Calculate [tex]\(\sin(\theta/2)\)[/tex]
Using the half-angle formula:
[tex]\[ \sin(\theta/2) = \sqrt{\frac{1 - \cos \theta}{2}} \][/tex]
[tex]\[ \sin(\theta/2) = \sqrt{\frac{1 - (-0.9863939238321439)}{2}} = \sqrt{\frac{1 + 0.9863939238321439}{2}} = \sqrt{\frac{1.9863939238321439}{2}} = 0.9965926760297167 \][/tex]
Step 7: Calculate [tex]\(\cos(\theta/2)\)[/tex]
Using the half-angle formula:
[tex]\[ \cos(\theta/2) = -\sqrt{\frac{1 + \cos \theta}{2}} \][/tex]
Since [tex]\(\theta/2\)[/tex] will be in the first or second quadrant and we should consider it negative due to [tex]\(\sec \theta < 0\)[/tex],
[tex]\[ \cos(\theta/2) = -\sqrt{\frac{1 + (-0.9863939238321439)}{2}} = -\sqrt{\frac{1 - 0.9863939238321439}{2}} = -\sqrt{\frac{0.013606076167856063}{2}} = -0.08248053154489289 \][/tex]
In summary:
(a) [tex]\(\sin(2\theta) = -0.32432432432432273\)[/tex]
(b) [tex]\(\cos(2\theta) = 0.9459459459459465\)[/tex]
(c) [tex]\(\sin(\theta/2) = 0.9965926760297167\)[/tex]
(d) [tex]\(\cos(\theta/2) = -0.08248053154489289\)[/tex]
Step 1: Find [tex]\(\tan \theta\)[/tex]
Since [tex]\(\cot \theta = -6\)[/tex], we know that:
[tex]\[ \cot \theta = \frac{1}{\tan \theta} \implies \tan \theta = \frac{1}{\cot \theta} = \frac{1}{-6} = -\frac{1}{6} \][/tex]
Therefore,
[tex]\[ \tan \theta = -0.16666666666666666 \][/tex]
Step 2: Find [tex]\(\sec \theta\)[/tex] and [tex]\(\cos \theta\)[/tex]
Given [tex]\(\sec \theta < 0\)[/tex], [tex]\(\theta\)[/tex] must be in the range where cosine is negative. This condition places [tex]\(\theta\)[/tex] in the second or third quadrant. Since we also need to satisfy the Pythagorean identity:
[tex]\[ \tan^2 \theta + 1 = \sec^2 \theta \][/tex]
we can calculate:
[tex]\[ (-0.16666666666666666)^2 + 1 = \sec^2 \theta \implies \sec^2 \theta = \frac{1}{36} + 1 = \frac{1}{36} + \frac{36}{36} = \frac{37}{36} = 1.027777777777778 \][/tex]
[tex]\[ \sec \theta = -\sqrt{1.027777777777778} = -1.0137937550497031 \][/tex]
Since [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex],
[tex]\[ \cos \theta = -1.0137937550497031^{-1} = -0.9863939238321439 \][/tex]
Step 3: Find [tex]\(\sin \theta\)[/tex]
Using the identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \implies \sin^2 \theta = 1 - \cos^2 \theta \][/tex]
[tex]\[ \sin^2 \theta = 1 - (-0.9863939238321439)^2 = 1 - 0.9730141283301396 = 0.026985871669860395 \][/tex]
[tex]\[ \sin \theta = \sqrt{0.026985871669860395} = 0.16439898730535646 \][/tex]
Step 4: Calculate [tex]\(\sin(2\theta)\)[/tex]
Using the double-angle formula for sine:
[tex]\[ \sin(2\theta) = 2 \sin \theta \cos \theta \][/tex]
[tex]\[ \sin(2\theta) = 2 \times 0.16439898730535646 \times -0.9863939238321439 = -0.32432432432432273 \][/tex]
Step 5: Calculate [tex]\(\cos(2\theta)\)[/tex]
Using the double-angle formula for cosine:
[tex]\[ \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \][/tex]
[tex]\[ \cos(2\theta) = (-0.9863939238321439)^2 - (0.16439898730535646)^2 = 0.9730141283301396 - 0.027068136054193123 = 0.9459459459459465 \][/tex]
Step 6: Calculate [tex]\(\sin(\theta/2)\)[/tex]
Using the half-angle formula:
[tex]\[ \sin(\theta/2) = \sqrt{\frac{1 - \cos \theta}{2}} \][/tex]
[tex]\[ \sin(\theta/2) = \sqrt{\frac{1 - (-0.9863939238321439)}{2}} = \sqrt{\frac{1 + 0.9863939238321439}{2}} = \sqrt{\frac{1.9863939238321439}{2}} = 0.9965926760297167 \][/tex]
Step 7: Calculate [tex]\(\cos(\theta/2)\)[/tex]
Using the half-angle formula:
[tex]\[ \cos(\theta/2) = -\sqrt{\frac{1 + \cos \theta}{2}} \][/tex]
Since [tex]\(\theta/2\)[/tex] will be in the first or second quadrant and we should consider it negative due to [tex]\(\sec \theta < 0\)[/tex],
[tex]\[ \cos(\theta/2) = -\sqrt{\frac{1 + (-0.9863939238321439)}{2}} = -\sqrt{\frac{1 - 0.9863939238321439}{2}} = -\sqrt{\frac{0.013606076167856063}{2}} = -0.08248053154489289 \][/tex]
In summary:
(a) [tex]\(\sin(2\theta) = -0.32432432432432273\)[/tex]
(b) [tex]\(\cos(2\theta) = 0.9459459459459465\)[/tex]
(c) [tex]\(\sin(\theta/2) = 0.9965926760297167\)[/tex]
(d) [tex]\(\cos(\theta/2) = -0.08248053154489289\)[/tex]
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