Functions+Notes+_updated_.pdf

39 let c be the circle with centre at the origin and

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Definition 2.39. Let C be the circle with centre at the origin and radius 1; its equation is x 2 + y 2 = 1. Let A be the point (1 , 0) on C . For any real number t , let P t the point on C at distance | t | from A , measured along C in the counterclockwise direction if t > 0, and the clockwise direction if t < 0. The cosine of t (abbreviated cos t ) and the sine of t (abbreviated sin t ) are the x -coordinate and y -coordinate of the point P t . Example 2.40. Examining the coordinates of P 0 = A , P π/ 2 , P π , and P - π/ 2 = P 3 π/ 2 in the figure below, we obtain the values of cosine and sine of 0, π/ 2, π , and 3 π/ 2. cos 0 = 1 cos π 2 = 0 cos π = - 1 cos 3 π 2 = 0 sin 0 = 0 sin π 2 = 1 sin π = 0 sin 3 π 2 = - 1
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50 J. S´ anchez-Ortega Convention. If θ is any real number, sin θ means the sine of the angle whose radian measure is θ . For example, the expression sin 3 implies that we are dealing with an angle of 3 rad. When finding a calculator approximation to this number, we must remember to set our calculator in radian mode, and then we obtain sin 3 0 . 14112 If we want to know the sine of the angle 3 we would write sin 3 and, with our calculator in degree mode, we find that sin 3 0 . 05234 Some Special Angles The exact trigonometric ratios for certain angles like, for example, π/ 4, π/ 3 and π/ 6 can be read from the triangles below. More precisely:
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2. Functions 51 We can also find the sine and cosine of π/ 4 determining the coordinates of the point P π 4 on the circle x 2 + y 2 = 1. The point P π 4 lies in the first quadrant on the line y = x . To find its coordi- nates, substitute y = x into the equation x 2 + y 2 = 1 of the circle, obtaining 2 x 2 = 1. Thus x = y = 1 / 2 and cos π 4 = 1 2 = 2 2 sin π 4 = 1 2 = 2 2 Concerning to π/ 3 notice that the point P π 3 and the points O (0 , 0) and A (1 , 0) are the vertices of an equilateral triangle with edge length 1. Thus P π 3 has x -coordinate 1 / 2 and y -coordinate p 1 - (1 / 2) 2 = 3 / 2, and cos π 3 = 1 2 sin π 3 = 3 2
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52 J. S´ anchez-Ortega The signs of the trigonometric functions for angles in each of the four quad- rants can be remembered by means of the rule ‘ A ll S tudents T ake C alculus” shown in the figure below. All three are positive in the first quadrant, marked A . Of the three, only sin is positive in the second quadrant, S , only tangent in the third quadrant T , and only cosine in the fourth quadrant C . Trigonometric Identities From the definitions of the trigonometric functions we have that tan θ = sin θ cos θ Many important properties of cos θ and sin θ follow from the fact that they are coordinates of the point (cos θ, sin θ ) on the unit circle C with equation x 2 + y 2 = 1. From here, we can derive already two identities: 1. The range of cosine and sine. - 1 cos θ 1 - 1 sin θ 1 2. The Pythagorean identity.
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