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Part One
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Figure 10-1 Illustration for Question and Answer 1-8 It doesn t matter which way we go when we want to determine the straight-line distance between two points Therefore, dst = dts
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Question 1-8
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In Fig 10-1, what do the expressions x and y mean What s the straight-line distance d between the two points, based on the values of x and y
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Answer 1-8
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We read x as delta x, which means the difference in x We read y as delta y, which means the difference in y The straight-line distance d between the points can be found by squaring x and y individually, adding the squares, and then taking the nonnegative square root of the result, getting d = ( x2 + y2)1/2
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Question 1-9
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Suppose we want to find the midpoint of a line segment connecting two known points in the Cartesian xy plane How can we do this
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Answer 1-9
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We average the x coordinates of the endpoints to get the x coordinate of the midpoint, and we average the y coordinates of the endpoints to get the y coordinate of the midpoint
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Review Questions and Answers Question 1-10
Once again, imagine two points S and T in the Cartesian plane with the coordinates S = (xs,ys) and T = (xt,yt) What are the coordinates of the point B that bisects the line segment connecting S and T
Answer 1-10
The point B is the midpoint of the line segment When we follow the procedure described in Answer 1-9, we obtain the coordinates (xb,yb) of point B as (xb,yb) = [(xs + xt)/2,( ys + yt)/2]
2
Question 2-1
What is a radian
Answer 2-1
A radian is the standard unit of angular measure in mathematics If we have two rays pointing out from the center of a circle, and those rays intersect the circle at the endpoints of an arc whose length is equal to the circle s radius, then the smaller (acute) angle between the rays measures one radian (1 rad )
Question 2-2
How many radians are there in a full circle In 1/4 of a circle In 1/2 of a circle In 3/4 of a circle
Answer 2-2
There are 2p rad in a full circle Therefore, 1/4 of a circle is p/2 rad, 1/2 of a circle is p rad, and 3/4 of a circle is 3p/2 rad
Question 2-3
Suppose we have an angle whose radian measure is 7p/6 What fraction of a complete circular rotation does this represent
Answer 2-3
Remember that an angle of 2p represents a full rotation The quantity p/6 is 1/12 of 2p, so an angle of p/6 represents 1/12 of a rotation Therefore, an angle of 7p/6 represents 7/12 of a rotation
Part One
Figure 10-2 Illustration for Questions and Answers
2-4 through 2-9 Each axis division represents 1/4 unit
Question 2-4
In Fig 10-2, the gray circle is a graph of the equation x2 + y2 = 1 The point (x0,y0) lies on this circle A ray from the origin through (x0,y0) subtends an angle q going counterclockwise from the positive x axis How can we define the sine of the angle q
Answer 2-4
The sine of q as shown in Fig 10-2 is equal to y0 Mathematically, we write this as sin q = y0
Question 2-5
How can we define the cosine of the angle q in Fig 10-2
Answer 2-5
The cosine of q is equal to x0 Mathematically, we write this as cos q = x0
Question 2-6
How can we define the tangent of the angle q in Fig 10-2
Review Questions and Answers Answer 2-6
The tangent of q is equal to y0 divided by x0, as long as x0 is nonzero If x0 = 0, then the tangent of the angle is not defined Mathematically, we have tan q = y0 /x0 x0 0 The double-headed, double-shafted arrow ( ) is the logical equivalence symbol It translates to the words if and only if We can also define the tangent as tan q = sin q /cos q cos q 0
Question 2-7
How can we define the cosecant of the angle q in Fig 10-2
Answer 2-7
The cosecant of q is equal to the reciprocal of y0, as long as y0 is nonzero If y0 = 0, then the cosecant is not defined Mathematically, we have csc q = 1/y0 y0 0 We can also define the cosecant as csc q = 1/sin q sin q 0
Question 2-8
How can we define the secant of the angle q in Fig 10-2
Answer 2-8
The secant of q is equal to the reciprocal of x0, as long as x0 is nonzero If x0 = 0, then the secant is not defined Mathematically, we have sec q = 1/x0 x0 0 We can also define the secant as sec q = 1/cos q cos q 0
Question 2-9
How can we define the cotangent of the angle q in Fig 10-2
Answer 2-9
The cotangent of q is equal to x0 divided by y0, as long as y0 is nonzero If y0 = 0, then the cotangent is not defined Mathematically, we have cot q = x0/y0 y0 0
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