Textbook Question
Graph two periods of the given tangent function. y = tan(x − π/4)
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2. Trigonometric Functions on Right Triangles
Solving Right Triangles
3:13 minutesProblem 57
Textbook Question
In Exercises 55–58, use a graph to solve each equation for -2π ≤ x ≤ 2π. csc x = 1
Verified step by step guidance1
Recall that the cosecant function is the reciprocal of the sine function, so \(\csc x = \frac{1}{\sin x}\). Therefore, the equation \(\csc x = 1\) can be rewritten as \(\frac{1}{\sin x} = 1\).
From the equation \(\frac{1}{\sin x} = 1\), multiply both sides by \(\sin x\) (noting \(\sin x \neq 0\)) to get \(1 = \sin x\).
Now, solve the equation \(\sin x = 1\) for \(x\) in the interval \(-2\pi \leq x \leq 2\pi\) by identifying where the sine function reaches the value 1 on its graph.
Recall that \(\sin x = 1\) at \(x = \frac{\pi}{2} + 2k\pi\) for any integer \(k\). Find all such \(x\) values within the given interval by substituting integer values for \(k\).
List all solutions found in the interval \(-2\pi \leq x \leq 2\pi\) as the final answer to the equation \(\csc x = 1\).
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Understanding the Cosecant Function
The cosecant function, csc x, is the reciprocal of the sine function, defined as csc x = 1/sin x. It is undefined where sin x = 0, and its values correspond to the reciprocal of sine values. Recognizing this relationship helps in solving equations involving csc x.
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Graphs of Secant and Cosecant Functions
Graphing Trigonometric Functions
Graphing csc x involves plotting the reciprocal of the sine curve, which has vertical asymptotes where sine is zero. Understanding the shape and key points of the csc x graph allows one to visually identify solutions to equations like csc x = 1 within a given interval.
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Solving Trigonometric Equations on a Given Interval
Solving csc x = 1 over -2π ≤ x ≤ 2π requires finding all x-values where csc x equals 1 within this domain. This involves identifying corresponding sine values (sin x = 1) and considering the periodicity of the sine and cosecant functions to list all valid solutions.
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