# Chapter 6 -...

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Unformatted text preview: Chapter 6:  Trigonometric Functions of Angles    Trigonometric Functions of Acute Angles  (6.1)      Vocabulary:  An angle is a figure formed by two rays with a common end point.  The  common end point is called the vertex.                                    < = angel = θ = < ABC             One measurement of angle is degrees.  A full rotation is 360 degrees.  The degree is  sub‐divided into smaller units.  The minute (1/60th of a degree)  and second (1/60th  of a minute)      Acute angles have measure    00 < θ < 900      Ex:                      Obtuse angels have measure   900 < θ < 1800     Ex:                      Definition:  Let θ  be an acute angle placed in a right triangle.  The basic trig  functions are defines as follows:                          You can remember the definition of the basic trigonometric functions by  remembering SOHCAHTOA      Reciprocal Identities can be defined from the basic trig functions as follows:                              Ex:   ΔABC  is a right triangle with AC = 3,  BC = 2  and <C = 900    Find the six trigonometric functions:                    Calculate  sin (250) , cos (250)  and tan (250) using degree mode in your calculator.              Two well known triangles from Geometry:      300‐600 Right Triangle    Theorem:  In a 300‐600  right triangle, the length of the opposite side to the 300 angel  is half of the length of the hypotenuse.                       450‐450 Right Triangle    Theorem:  In a 450‐450 right triangle, the length of the opposite and the adjacent  sides are equal.                                  Let’s now make a table with the trig functions with all the special angels where you  can express them as an exact answer (without calculator).   You have to memorize  this table asap.                                                                                                                                        Prove:  cos(600) = cos2(300) – sin2(300)                                Algebra and the Trigonometric Functions (6.2)    Notation:      1)   sin(θ ) = said as sin e of the angel θ = sinθ     In this case θ is the input and sinθ is the output     Note:   sinθ is a value ; For example, sin(300) = ½    2)  When you are multiplying with two trigonometric quantities, parenthesis is often  omitted.    For example,    (sinθ )(cosθ ) = sinθ cosθ Similarly (sinθ )(sinθ ) = sin 2θ   (sinθ )n = sin n θ (n ≠ -1)   The convention stated above applies to the other five trig functions.      Useful Hint:  Treat  sinθ ,cosθ  or any of the other four trig expression as you treat  variables (like x and y) in Algebra while simplifying trigonometric quantities.      Ex:  Simplify the following expressions:    1)  10 sinθ cosθ + 4 sinθ cosθ − 16 sinθ cosθ =                     2)   ( 3 sinθ − 4 cosθ )( 2 sinθ + cosθ ) =               Factor:   tan 2 θ − 2 tanθ − 15 =                            Basic Right Angle Identities:      1) sin 2θ + cos 2 θ =     2)   tanθ =     3)   sin( 90 − θ ) =     4)   cos( 90 − θ ) =     cscθ = 5)   secθ =       Proof:                              cotθ =  Ex:  Given  cos A = functions.                                        Ex:  Simplify                              4 , and A is an acute angel, find the other five trigonometric  5 sin 2 θ − cos 2 θ =  sinθ − cosθ 1 − 2 cos 2 x Ex:   cot x +   sin x cos x           Right Triangle Applications  (6.3):        We are going to apply the knowledge obtained in the sections 6.1 and 6.2 in solving  some basic problems involving right triangles.      Ex:  In a 300‐600 right triangle, the hypotenuse is 60 cm.  Find the measure of the  other two sides                                            Ex:  In a 450‐450 right triangle, the opposite side has length 9 inches.  Find the  measure of the other two sides.                                A trigonometric formula for the area of a triangle with side lengths a and b and the  angle in between θ  :    1 1 Formula : Area = ( base )( height ) sinθ = ab sinθ   2 2   Proof:                                            Ex:  Find the area of a triangle with side lengths a = 5, b = 9 and the angle in between  θ  = 600                              Ex:  Find the area of a rectangular nonagon that is inscribed in a circle of radius 6  cm.                                                      Ex:  A certain sprinkler will water ground out to a distance of 25 feet.  If the  sprinkler head is set to rotate back and forth through an angel of 600, determine the  area in square feet that is being irrigated.                              Definition:  The angle of elevation is the angle formed by the horizontal and the  line of sight to an object above the observer.                                        The angle of depression is the angel formed by the horizontal and the line of sight  to an object below the observer.                                              Ex:  From a point on the ground level, you measure the angle of elevation to the top  of the mountain to be 380.  Then you walk 200m further away from the mountain  and find that the angle of elevation is now 200.  Find the height of the mountain to  the nearest meter.                                                                                      Trigonometric Function of Angles (6.4)    You need to bring a copy of a Unit Circle to lecture today!  Download it from  my web site.    In this section we will not be restricted to acute angle  ( 00 < θ < 900 ) trigonometry  only.  We will expand our definitions to include angles of any size.   Remember the definition of angle formed by two rays (sec 6.1).  Let’s fix one side of  the angle (the initial side) as shown below and rotate through an angle θ  to get to  the terminal side as shown below.                      By convention, the counterclockwise rotation yields a positive angle measure θ , and  the clockwise rotation yields a negative angle measure (‐θ ) as shown below.                        Standard Position:  Initial side is always aligned with the positive x‐axis of the Unit  Circle as shown below:                    After the rotation of angle θ  we land on the terminal side with the coordinate (x, y).    We can define the six trig functions in terms of x and y as follows:                                  (x, y) = coordinate of the terminal side = (cosθ , sinθ )    Since (x, y) lies on the Unit Circle, it must satisfy the equation of the circle with  radius 1 which is     x 2 + y 2 = 1    check:                By fixing the initial side in the standard position and by rotating the terminal side by  an angel measure 00<θ <3600 , we are able to find trigonometric values of any angle  θ .  Visualize the rotation to measure angle of any size on an Unit Circle as  demonstrated below:                Now let’s fill out the table below by consulting at the Unit Circle:                                                                                                                                Let’s discuss how the Unit Circle works since you have to be an expert on this very  soon!!        What if the angle measurement is not always falling on the axis as shown on the  table above. Now we will find a procedure to find trigonometric values of angles that  do not necessarily fall on the x or y axis.    Definition:  You can remember the sign of the trigonometric function value by  remembering All Students Take Calculus as the quadrant changes as shown below:      Definition:  The reference angle associated with an angel θ  in standard position is  the acute angle (with positive measure) formed by the x­axis and the terminal side  of the angle θ .  Ex:  Find the reference angle for 2250              Once we have established how to find the reference angle, following is the  procedure to calculate trig function value of any angel.        Ex:  Find the value of  sin 2250  after figuring out the corresponding reference angel.    a) Sketch the angel                          b) Determine the reference angel            c) Evaluate the trig function of the reference angel.            d) Affix the appropriate sign depending on which quadrant the angle lies in.                Ex:  Use the same steps as described in the example above to find the  following quantities:      cos1500              sin1500                     sin(‐1500)                                                      cos(‐1500)                                          Ex:  Find sin(4050) , cos(4050) and tan (4050)                                            Ex: Find the area of a triangle with sides 5 cm, 7cm and the angle between them is  1200.                                              Trigonometric Identities (6.5)      Basic Identities that you must memorize:    1) sin 2θ + cos 2 θ = 1 Pr oof : 2) tanθ = sinθ cosθ 3) secθ = 1 cosθ cscθ = 1 sinθ 4 ) 1 + tan 2θ = sec2 θ Proof : 5) 1 + cot 2θ = csc2 θ Proof :                 These identities are true for all angles θ   cotθ = 1   tanθ Ex:  Given  cosθ =                                                                                       3 , 2700 < θ < 3600 ( Q IV)     Find all the other five trig functions.  4   Ex:  Given  sinθ = −3u , 1800 < θ < 2700    Find all the other five trig functions.                                                                                          Prove the following identities:    Ex:   tan β sin β = sec β − cos β                                             1 − tan 2 α Ex:  Show  cos2 α − sin 2 α =   1 + tan 2 α                                     Ex:  Show  cosθ 1 + sinθ   = 1 − sinθ cosθ     Note:  Since this example is already in terms of sinθ  and cosθ , you will have to use  the algebraic technique of rationalizing the denominator.  Then you will be able to  use the basic identities.                                                                              Ex:   secθ − cscθ tanθ − 1   = secθ + cscθ tanθ + 1   Note:  In this example, you may have to work with both sides of the equation .                                                                                                                                    ...
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## This note was uploaded on 08/09/2011 for the course MATH 3 taught by Professor Staff during the Spring '08 term at University of California, Santa Cruz.

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