1.Secy/coty + tany)2.tanycoty)(tany+ coty)3.( Siny + cosy)(secy + cscy)4.tany(cosy - cscy)5.Siny +cosy)(secy +cscy)6.(siny-cosy)^2 9.tan165)10.sin225)12.cos165)

Question

These are multiple questions that I need help with. Numbers 1.( Ask to simplify) 2-6 (ask to multiply and simplify) and 9-14 ask to find angle sum and difference. 

 

Note: I'm also looking for a tutor.... I have a budget of 35/hr for 2hrs.. I Live in the santa Monica area.. I need a Trigonometry tutor for over the summer.... We will work to get me ahead as I will take the course next year... If you want to discuss higher price pleased email me so I can discuss matter with my parents but no higher than 50/ per hour 

Answer 1

1) sec y/cot y + tan y = 1/cos y*sin y/cos y + sin y/cos y = sin y /cos y^2 + sin y/cos y = (sin y+ sin y*cos y)/cos y^2 = sin y (1+cos y)/cos y^2

 

2)  tany coty = 1, so tan y + cot y = sin y/cos y + cos y/sin y = (cos y^2 + sin y ^2 )/ (sin y cos y) = 1 /(sin y cos y)

 

3) (sin y + cos y ) (sec y + csc y ) =  sin y sec y + sin y csc y + cos y sec y + cos y + csc y   using FOIL

                                                    =     tan   y     +     1         +      1            +   cot y

                                                     =  tan y  + cot y + 2

 

                                                     =  2 + 1/(sin y cos y)   <-- from previous exercise

 

                                                     = (2 sin y cos y + 1)/ (sin y cos y)

 

                                                       = (2 sin 2y + 1) / (sin y cos y)

 

4) tan y ( cos y - csc y) = tan y ( cos y - 1/sin y) = sin y/cos y ( cos y sin y - sin y)/siny

 

                                                                          =  (cos y sin y - sin y )/cos y 

 

                                                                           = sin y - tan y

 

5)   IS EXACTLY THE SAME AS #3

 

6)  (sin y - cos y) = sin y^2 - 2 sin y cos y + cos y ^2 =  1 - 2 sin y cos y = 1 - sin 2y

 

 

mnemonic device: "ALL STUDENTS TAKE CALCULUS"

        1st quadrant: A --> all trig functions positive

        2nd quadrant: S--> sine positive only

        3rd quadrant: T --> tangent function positive only

        4th quadrant: C --> cosine function positive only

 

9) tan 165 is in the 2nd quadrant where tangent is negative. This is 15 above  the negative side of the x axis.

     

      tan(180-15) = [tan(180)-tan(15)]/ [1 - tan(180)tan(15)] = [ 0 - tan(15)] /  [1 + 0 * tan(15)] = -tan15 = -02679...

 

 

10) sin 225  is in the 3rd quadrant where sine is negative

 

    It is 45 degrees below the negative side of the x axis.

 

   sine (225) = sin(180 + 45) = sin(180)cos(45) + cos(180)sin(45) 

                                              =   0 * cos(45) + -1(sin(45) =

                                                                -sin(45) = -rad2/2 where rad2 = square-root(2)

 

12) cos(165) is in the second quadrant where cosine is negative.

 

    It is 15 degrees above the negative x-axis

 

   cos(165) = cos(180-15) = cos(180)cos(15) - sin(180)sin(15)

                                                    -1  cos(15  -  0 * sin(15)

 

                                                      = -cos(15) = -.9659258262891