Angle of elevation-trigonometry

trigonometric angles values and trigonometric functions of angles in standard position
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Prof.EvanBaros,United Kingdom,Teacher
Published Date:26-07-2017
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Angles MATH 1510 An angle\AOB consists of two rays R and R with a 1 2 Lili Shen common vertex O. Angle Measure A R 2   R 1 O Trigonometry  B of Right Triangles  B   R 1 R 2 O A Positive angle Negative angle We often interpret an angle as a rotation of the ray R 1 (called the initial side) onto R (called the terminal side). If 2 the rotation is counterclockwise, the angle is considered positive, and if the rotation is clockwise, the angle is considered negative.Angle measure MATH 1510 Lili Shen Angle The measure of an angle is the amount of rotation about the Measure Trigonometry vertex required to move R onto R . One unit of 1 2 of Right Triangles measurement for angles is the degree:  An angle of measure 1 degree, written as 1 , is formed 1 by rotating the initial side of a complete revolution. 360 In calculus and other branches of mathematics a more natural method of measuring angles is used: radian measure.Angle measure MATH 1510 Definition Lili Shen If a circle of radius 1 is drawn with the vertex of an angle at Angle Measure its center, then the measure of this angle in radians Trigonometry (abbreviated rad) is the length of the arc that subtends the of Right Triangles angle:   Radian measure of  1Angle measure MATH 1510 Lili Shen Angle Measure Trigonometry The circumference of the circle of radius 1 is 2, so a of Right Triangles complete revolution has measure 2 rad, a straight angle  has measure rad, and a right angle has measure rad. 2 Similarly, an angle that is subtended by an arc of length 2 along the unit circle has radian measure 2.Angle measure MATH 1510 Lili Shen From the definition one easily knows the relationship Angle between degrees and radians: Measure  Trigonometry 180 = rad; of Right    Triangles 180 1 rad = ;    1 = rad. 180 For example,   60 = rad, 3   rad = 30 . 6Angle measure MATH 1510 Lili Shen Angle Measure  We often use a phrase such as “a 30 angle” to mean an Trigonometry  of Right angle whose measure is 30 . Triangles   Also, for an angle we write = 30 or = to mean the 6   measure of is 30 or rad. 6 When no unit is given, the angle is always assumed to be measured in radians.Angles in standard position MATH 1510 Lili Shen Angle Measure Trigonometry of Right An angle is in standard position if it is drawn in the xy-plane Triangles with its vertex at the origin and its initial side on the positive x-axis. Two angles in standard position are coterminal if their sides coincide.Coterminal angles MATH 1510 Lili Shen Angle Measure Trigonometry of Right Example Triangles  (1) Find angles that are coterminal with the angle = 30 in standard position.  (2) Find angles that are coterminal with the angle = in 3 standard position.Coterminal angles MATH 1510 Lili Shen Angle Measure Trigonometry of Right Solution. Triangles  (1) (30+ 360k) , where k is any integer.  (2) + 2k, where k is any integer. 3Length of a circular arc MATH 1510 Lili Shen Proposition Angle Measure In a circle of radius r the length s of an arc that subtends a Trigonometry central angle of radians is of Right Triangles s = r:  s= r rLength of a circular arc MATH 1510 Lili Shen Angle Measure Trigonometry Proof. of Right Triangles The circumference of a circle of radius r is 2r. Hence the length s of an arc that subtends a central angle of radians is  s = 2r = r: 2Length of a circular arc MATH 1510 Lili Shen Angle Measure Trigonometry of Right Therefore, the radian measure of Triangles s  = r is the number of “radiuses” that can fit in the arc that subtends; hence the term radian.Length of a circular arc MATH 1510 Lili Shen Angle Measure Trigonometry of Right Example Triangles (1) Find the length of an arc of a circle with radius 10 m  that subtends a central angle of 30 . (2) A central angle in a circle of radius 4 m is subtended by an arc of length 6 m. Find the measure of.Length of a circular arc MATH 1510 Lili Shen Angle Measure Solution. Trigonometry   of Right (1) The angle = 30 = , and thus the length of the arc is Triangles 6  5 s = r = 10 = : 6 3 s 6 3 (2)  = = = . r 4 2Area of a circular sector MATH 1510 Lili Shen Angle Measure Trigonometry of Right Proposition Triangles In a circle of radius r the area A of a sector with a central angle radians is 1 2 A = r : 2Area of a circular sector MATH 1510 Lili Shen Angle Measure Trigonometry Proof. of Right 2 Triangles The area of a circle of radius r isr . Hence the area A of a sector with a central angle radians is  1 2 2 A =r  = r : 2 2Area of a circular sector MATH 1510 Lili Shen Angle Measure Trigonometry of Right Triangles Example  Find the area of a sector of a circle with central angle 60 if the radius of the circle is 3 m.Area of a circular sector MATH 1510 Lili Shen Angle Measure Trigonometry of Right Triangles Solution. 1 1  3 2 2 2 A = r  =  3  = m : 2 2 3 2Circular motion MATH 1510 Lili Shen Suppose a point moves along a circle as shown below: Angle Measure Trigonometry of Right Triangles s   r There are two ways to describe the motion of the point: linear speed and angular speed.Circular motion MATH 1510 Lili Shen Angle Measure Linear speed and angular speed Trigonometry of Right Suppose a point moves along a circle of radiusr and the ray Triangles from the center of the circle to the point traverses radians in time t. Let s = r be the distance the point travels in time t. Then s the linear speed is given by v = , t  the angular speed is given by = . t

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