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Part One
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We can also define the cotangent as cot q = 1/tan q tan q 0 or as cot q = cos q /sin q sin q 0
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What are the Pythagorean identities for trigonometric functions Which, if any, of these should be memorized
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Answer 2-10
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The Pythagorean identities are the three formulas sin2 q + cos2 q = 1 sec2 q tan2 q = 1 csc2 q cot2 q = 1 The first of these is worth memorizing, because it comes up quite often in applied mathematics and engineering The second and third identities can be derived from the first one
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Question 3-1
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How are variables and points portrayed on the polar-coordinate plane
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Answer 3-1
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The independent variable is rendered as a direction angle q, expressed counterclockwise from a reference axis This reference axis normally goes outward from the origin toward the right (or due east ), in the same direction as the positive x axis in the Cartesian xy plane The dependent variable is rendered as a radius r, expressed as the straight-line distance from the origin Points in the plane are expressed as ordered pairs of the form (q,r), as shown in Fig 10-3 In some texts, the ordered pair is written as (r,q)
Question 3-2
Can a point in polar coordinates have a negative direction angle, or an angle that represents a full rotation or more
Answer 3-2
Yes If q < 0, it represents clockwise rotation from the reference axis If q 2p, it represents at least one complete counterclockwise rotation from the reference axis
Question 3-3
Can a point in polar coordinates have a negative radius
Review Questions and Answers
p /2
Angle is q
Radius is r (r, q)
3p /2
Figure 10-3 Illustration for Question and Answer 3-1
Answer 3-3
Yes If r < 0, we can multiply r by 1 so it becomes positive, and then add or subtract p to or from the direction angle, keeping it within the preferred range 0 q < 2p
Question 3-4
How we can we portray a relation or function in polar coordinates when the independent variable is q and the dependent variable is r
Answer 3-4
We can write down an equation with r on the left-hand side and the name of the function followed by q in parentheses on the right-hand side For example, if our function is g, we write r = g (q) and read it as r equals g of q
Question 3-5
If we set the polar-coordinate angle equal to a constant, say k, what graph do we get
Answer 3-5
The graph is a straight line passing through the origin The line appears at an angle of k radians with respect to the reference axis
Question 3-6
If we set the polar-coordinate radius equal to a constant, say m, what graph do we get
Part One Answer 3-6
The graph a circle centered at the origin, so that every point on the circle is m units from the origin
Question 3-7
Suppose we have a point (q,r) in polar coordinates How can we convert this to coordinates in the Cartesian xy plane
Answer 3-7
We can convert the polar point (q,r) to Cartesian (x,y) using the formulas x = r cos q and y = r sin q
Question 3-8
Suppose we have a point (x,y) in the Cartesian plane What s the polar radius r of this point
Answer 3-8
The polar radius of a point is its distance from the origin We can use the formula for the distance of a point from the origin to find that r = (x2 + y2)1/2 This gives us a positive value for the radius, which is preferred
Question 3-9
Suppose we have a point (x,y) in the Cartesian plane What s the polar angle q of this point
Answer 3-9
This problem breaks down into following nine cases, depending on where in the Cartesian plane our point (x,y) lies: If x = 0 and y = 0, then q = 0 by default If x > 0 and y = 0, then q = 0 If x > 0 and y > 0, then q = Arctan ( y /x) If x = 0 and y > 0, then q = p /2 If x < 0 and y > 0, then q = p + Arctan ( y /x) If x < 0 and y = 0, then q = p If x < 0 and y < 0, then q = p + Arctan ( y /x) If x = 0 and y < 0, then q = 3p /2 If x > 0 and y < 0, then q = 2p + Arctan ( y /x)
If we follow this process carefully, we always get an angle in the range 0 q < 2p, which is preferred
Review Questions and Answers Question 3-10
What does Arctan mean in the conversions listed in Answer 3-9
Answer 3-10
It stands for Arctangent That s the function that undoes the work of the trigonometric tangent function The domain of the Arctangent function is the entire set of real numbers The range is the open interval ( p /2,p /2) For any real number u within this interval, we have Arctan (tan u) = u Conversely, for any real number v, we have tan (Arctan v) = v
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