Explanation Given: The coordinates of a point ( 0.3965 , 0.918 ) The point is rotated about the origin to generate new points as shown in the Figure 23(A) The point is reflected about the x and y axes to generate new points as shown in the Figure 23(B) When transforming a given point, the following rules are to be followed. Rotation of a point about the origin When a point with coordinates ( x , y ) is rotated 90 o counterclockwise we obtain the new coordinates for the point ( − y , x ) When a point with coordinates ( x , y ) is rotated 90 o clockwise we obtain the new coordinates for the point ( y , − x ) When a point with coordinates ( x , y ) is rotated 180 o clockwise or counterclockwise we obtain the new coordinates for the point ( − x , − y ) Reflection of a point about the axes When a point with coordinates ( x , y ) is reflected about the x -axis we obtain the new coordinates for the point ( x , − y ) When a point with coordinates ( x , y ) is reflected about the y -axis we obtain the new coordinates for the point ( − x , y ) Reflection of a point about the origin When a point with coordinates ( x , y ) is reflected about the origin we obtain the new coordinates for the point ( − x , − y ) Let the given point be A . The following table shows the points generated when transforming the given point A ( 0.3965 , 0.918 ) No. Transformation New point Let the point be B Rotation 90 o counterclockwise. As per rule 1, the new coordinates are ( − y , x ) B ( − 0.918 , 0.3965 ) Let the point be C Rotation 90 o clockwise. As per rule 2, the new coordinates are ( y , − x ) C ( 0.918 , − 0.3965 ) Let the point be D Rotation 180 o clockwise or counterclockwise. As per rule 3, the new coordinates are ( − x , − y ) D ( − 0.3965 , − 0.918 ) Let the point be E Reflection about the x -axis. As per rule 4, the new coordinates are ( x , − y ) E ( 0.3965 , − 0.918 ) Let the point be F Reflection about the x -axis. As per rule 5, the new coordinates are ( − x , y ) F ( − 0.3965 , 0.918 ) Let the point be G Reflection about the origin. As per rule 6, the new coordinates are ( − x , − y ) G ( − 0.3965 , − 0.918 ) The following graphs shows the given point and the new points generated When transforming by rotation and reflection, the points lie on the same circle. By dropping perpendiculars from each point to the x-axis, and with the line to origin from the point, we can construct a right triangle from each of the points. For each of the triangles, the opposite is the y -coordinate of the corresponding point, adjacent is the x -coordinate of the corresponding point and the hypotenuse is the radius of the circle which can be determined by applying the Pythagorean Theorem for any one of the points. Formulas used: From basic trigonometry we have sin θ = Opposite Hypotenuse ; cos θ = Adjacent Hypotenuse ; tan θ = Opposite Adjacent The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Hypotenuse 2 = Opposite 2 + Adjacent 2 Hypotenuse = Opposite 2 + Adjacent 2 Calculations: Consider the point A, Angle for the point is θ Opposite = y − coordinate = 0.918 Adjacent = x − coordinate = 0.3965 Hypotenuse = Opposite 2 + Adjacent 2 = 0.3965 2 + 0.918 2 = 0.157 + 0.843 = 1 = 1 Hypotenuse = r a d i u s o f t h e c i r c l e = 1 Thus, for the angle corresponding to the point A , we have sin θ = Opposite Hypotenuse = 0.918 1 = 0.918 cos θ = Adjacent Hypotenuse = 0.3965 1 = 0.3965 tan θ = Opposite Adjacent = 0.918 0.3965 = 2.315 The same can be done for the other points. The following table shows the calculation of trigonometric values for all the points. No Point Values Trigonometric ratios sin θ = Opposite Hypotenuse cos θ = Adjacent Hypotenuse tan θ = Opposite Adjacent 1 B ( − 0

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Author: Rogawski

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Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.

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Chapter 1.4, Problem 29E

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The trigonometric values for the angles corresponding to the given points in the unit circle

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The trigonometric values for the angles corresponding to the given points in the unit circle

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Chapter 1 Solutions