Textbook Question
Use the formula ω = θ/t to find the value of the missing variable.
θ = 3.871 radians, t = 21.47 sec
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3. Unit Circle
Trigonometric Functions on the Unit Circle
3:21 minutesProblem 66
Textbook Question
Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. See Example 6. √3 cot θ = - —— 3
Verified step by step guidance1
Start by writing down the given equation: \(\cot \theta = -\frac{\sqrt{3}}{3}\).
Recall that \(\cot \theta = \frac{1}{\tan \theta}\), so rewrite the equation as \(\frac{1}{\tan \theta} = -\frac{\sqrt{3}}{3}\), which implies \(\tan \theta = -\frac{3}{\sqrt{3}}\).
Simplify \(\tan \theta = -\frac{3}{\sqrt{3}}\) to get \(\tan \theta = -\sqrt{3}\).
Determine the reference angle where \(\tan \theta = \sqrt{3}\). From the unit circle or common angles, \(\tan 60^\circ = \sqrt{3}\), so the reference angle is \(60^\circ\).
Since \(\tan \theta\) is negative, find all angles in the interval \([0^\circ, 360^\circ)\) where tangent is negative. Tangent is negative in the second and fourth quadrants, so the solutions are \(\theta = 180^\circ - 60^\circ\) and \(\theta = 360^\circ - 60^\circ\).
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cotangent Function and Its Definition
Cotangent (cot θ) is the reciprocal of the tangent function, defined as cot θ = cos θ / sin θ. Understanding this relationship helps in converting cotangent values into sine and cosine ratios, which is essential for solving trigonometric equations.
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Solving Trigonometric Equations in a Given Interval
When solving for θ within [0°, 360°), it is important to find all angles that satisfy the equation. This involves considering the periodicity of trigonometric functions and identifying all solutions in the specified range, including those in different quadrants.
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Sign of Trigonometric Functions in Different Quadrants
The sign of cotangent depends on the signs of sine and cosine, which vary by quadrant. Cotangent is negative where sine and cosine have opposite signs, specifically in the second and fourth quadrants, guiding the selection of correct angle solutions.
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