Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Experience the convenience of finding accurate answers to your questions from knowledgeable professionals on our platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
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]
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
