Explanation Given: cos ( θ + π 2 ) sin ( θ + π 2 ) To express cos ( θ + π 2 ) and sin ( θ + π 2 ) in terms of cos θ and sin θ let us assume a point A ( a , b ) and let θ be the angle corresponding to the point A. An angle θ + π 2 represents the rotation of the point counterclockwise by 90 o to generate a new point B ( c , d ) . By calculating the trigonometric values for both of the points we can determine the relationship between them. 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 ) To calculate the trigonometric values, we need the values of opposite, adjacent and hypotenuse at each of the points. 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. From the figure we know that the radius of the circle is 1. Therefore, the hypotenuse = 1 Formula used: From basic trigonometry we have sin x = Opposite Hypotenuse ; cos x = Adjacent Hypotenuse 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: By rotation of the point A ( a , b ) counterclockwise by 90 o we obtain the point B ( c , d ) . When a point with coordinates ( x , y ) is rotated 90 o counterclockwise we obtain the new coordinates for the point ( − y , x ) . Therefore the coordinates of the point B is B ( − b , a ) ...

Loose-leaf Version for Calculus: Early Transcendentals Combo 3e & WebAssign for Calculus: Early Transcendentals 3e (Life of Edition)

3rd Edition

ISBN: 9781319019846

Author: Jon Rogawski, Colin Adams

Publisher: W. H. Freeman

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Question

Chapter 1.4, Problem 32E

To determine

The terms given trigonometric expression in terms of $\cos θ$ and $\sin θ$

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Chapter 1.4, Problem 31E

Chapter 1.4, Problem 33E

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