Handouts10 - Calculus of Trigonometric Functions

Handouts10 - Calculus of Trigonometric Functions - MATH0201...

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MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS Calculus of Trigonometric Functions Dr. WONG Chi Wing Department of Mathematics, HKU MATH0201 BASIC CALCULUS Trigonometric Functions Trigonometric Functions on the Real Line Some Useful Identities Differentiation of Trigonometric Functions Integration of Trigonometric Functions MATH0201 BASIC CALCULUS Trigonometric Functions Trigonometric Functions on the Real Line Angle An angle is formed by rotating a ray about its endpoint. I The endpoint of the ray is called the vertex . I The initial position of the ray is called the initial side . I The final position of the ray is called the terminal side . MATH0201 BASIC CALCULUS Trigonometric Functions Trigonometric Functions on the Real Line Definition 1 In the Cartesian plane, an angle is said to be in standard position if its vertex is at the origin and its initial side is along the positive x –axis. Angles can be represented by (real) numbers. I The direction of rotation may be counterclockwise or clockwise which will be considered to be positive and negative respectively. I Magnitudes are usually measured in degree or in radian .
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MATH0201 BASIC CALCULUS Trigonometric Functions Trigonometric Functions on the Real Line Definition 2 (Radian) A central angle subtended by an arc of length equal to the radius of the circle is said to have radian measure 1, written 1 rad . I Circumference has length 2 ± r , there are 2 ± rad in one revolution. I 360 ± = 2 ± rad or 180 ± = ± rad . Convention We measure angles in radians and hence the unit rad will be suppressed. MATH0201 BASIC CALCULUS Trigonometric Functions Trigonometric Functions on the Real Line Special Angles in Radians and in Degrees Radian 0 ± 6 ± 4 ± 3 ± 2 ± 2 ± Degree 0 30 45 60 90 180 360 MATH0201 BASIC CALCULUS Trigonometric Functions Trigonometric Functions on the Real Line Sine and Cosine of an Angle Consider the unit circle in a coordinate system centered at the origin. I Terminal side of an angle ² in standard position will pass through the circle at a point P . I Abscissa of P is called the cosine of ² , denoted by cos ² . I Ordinate of P is called the sine of ² , denoted by sin ² . MATH0201 BASIC CALCULUS Trigonometric Functions Trigonometric Functions on the Real Line Note that if ³ = ² + 2 k ± where k 2 Z , then cos ³ = cos ² and sin ³ = sin ²: cos ( 3 ± ) = cos ( 2 ± + ± ) = cos ± sin 7 ± 2 = sin ± 2 ± + 3 ± 2 ² = sin 3 ± 2 cos ² ± 4 = cos ± 2 ± + ² ± 4 ² = cos 7 ± 4 :
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MATH0201 BASIC CALCULUS Trigonometric Functions Trigonometric Functions on the Real Line Definition 3 (Sine and Cosine Functions) The function which sends every real number x to the sine of an
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This note was uploaded on 04/22/2011 for the course MATH 201 taught by Professor C.wong during the Spring '11 term at HKU.

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Handouts10 - Calculus of Trigonometric Functions - MATH0201...

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