Section 7-I Multiple - Angle and Product - Sum Formulas

 
 
I. General Procedure

        a) collect like terms on one side of equation

        b) Isolate trigonometric function

        c) Find all solutions in interval [0, 2] or [0, ]

        d) Find infinite solutions using general formular:

        
I - double-angle formulas:

        a) 

        b) 

                    

                   

        c) 

 II. Reduced power of trig. Fcts:

        a) 

        b) 

        c) 
III. Half-angle formulas

        a) 

        b) 

        c) 

 IV. Product-sum formulas

        a) 

        b) 

        c) 

        d)
V. Sum-product formula

        a) 

        b) 

        c) 

        d)
 

 VI. Sum and Diffrence Formulas:

    sin (u +/- v) = sin u cos v +/- cos u sin v

    cos (u +/- v) = cos u cos v -/+ sin u sin v

    tan (u +/- v) = (tan u +/- tan v) / (1 -/+ tan u tan v)

        e.g. sin (75o) = sin (45o + 30o)

        tan 50o = (tan 40o + tan 10o) / (1 - tan 40o tan 10o)

 

Section 7-II Law of Sines
 
 
I. Given any oblique triangle with sides a, b, and c,
    then the Law of Sines says:

              

 II. Solve an oblique triangle using Law of Sines
     for the remaining three parts if given:

            A. Two angles and any side (AAS or ASA)

 
III. Solve an oblique triangle using Law of Sines
     for the remaining three parts if given:

            B. Two sides and an angle opposite one of them (SSA)

                    1. If A is acute and h = b sin A:

                                (a) a < h, no triangle is possible.

                                (b) a = h or a > h, one triangle is possible.

                                (c) h < a< b, two triangles are possible.

                    2. If A is obtuse and h = b sin A:

                                (a) a  b, no triangle is possible.

                                (b) a > b, one triangle is possible.
IV. The area of any triangle equals one-half the product of the lengths
      of two sides times the sine of their included angles.

            Area = 1/2 ab sin C = 1/2 ac sin B = 1/2 bc sin A,

                            remember area is in square units
 
 
 
 
 
 
 
 
 
 

 

Section 7-III   Law of Cosines
 
I. If ABC is any oblique triangle with sides a, b, and c, then the Law of Cosines says:

        (a) 

        (b) 

        (c) 

 II. Solve an oblique triangle using Law Cosines for the remaining three parts if given:

        (a). Three sides (SSS)

        (b). Two sides and their included angles (SAS)
 
 

 
III. Given any triangle with sides of length a, b, and c, then the area of the triangle is:

        (Heron's Formula) Area =      

Section 7-IV Complex Number and Trigonometry / DeMoivre's Theorem
 
 
I. Representation of complex number graphically

            z = a + bi graph a (real number coefficent) on horizontal axis and
                                    b (imaginary number coefficent) on vertical axis
 
 
 

 
II Trigonometric Form of complex number :

            z = a + bi is z = 

                        (a) a = r cos                          (b) b = r sin 

                        (c) r =, r is called the modulus of z 

                        (d) tan  = b/a;  is called the argument of z
III. DeMoivre's Theorem: If z = r(cos  + isin  ), then any positive integer n,

             

 IV. Roots of complex numbers:

                For any positive integer n, z = r(cos  + isin ) has n distinct n th roots given by:

             

                      where k = 0, 1, 2, .... n-1