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sara_87
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Homework Statement
if i have a polar equation:
r=arctan(tan(x))
is that the same as:
r=x ??
sara_87 said:i am trying to draw the graphs but can't becuase some values for x give a math error.
so how would i draw the graph for r=arctan(tan(x))?
The equation r=arctan(tan(x)) and r=x explores the concept of equivalence in mathematics. It states that the polar coordinate r, which represents the distance from the origin, is equal to the tangent of the angle x when arctangent is applied, and also equal to the angle x itself.
This equation relates to the trigonometric functions of tangent and arctangent, which are inverse functions of each other. The arctangent function "undoes" the tangent function, resulting in the original angle x. Therefore, when r=arctan(tan(x)), the r value is equal to the angle x in radians.
One example is in navigation, where polar coordinates are used to determine the position of an object or person. The equation r=arctan(tan(x)) and r=x can be used to convert between Cartesian coordinates (x,y) and polar coordinates (r,θ). This can be useful in situations such as locating a missing person in a search and rescue mission.
The equation is true when x is within the range of values for which arctangent is defined, which is -π/2 to π/2 in radians. However, if x is outside of this range, the equation may not hold. For example, when x=π/2, r=π/2, but when x=3π/2, r=-π/2, which is not equal to x. Therefore, it is important to consider the range of x values when using this equation.
The equivalence can be proven using the properties of inverse trigonometric functions and the definition of polar coordinates. By applying the inverse tangent function (arctan) to both sides of the equation, we can show that arctan(tan(x))=x. Then, by using the definition of tangent (opposite/adjacent), we can see that r=arctan(tan(x)) is equal to the angle x in radians. Therefore, r=x, proving the equivalence.