Mathematics Magazine for Grades 1-12  

Grade 11

Theory:

The Magic Identity

Trigonometry is the art of doing algebra over the circle. So it is a mixture of algebra and geometry. The sine and cosine functions are just the coordinates of a point on the unit circle. This implies the most fundamental formula in trigonometry (which we will call here the magic identity)

cos²(θ) + sin²(θ) = 1

where θ is any real number (of course θ measures an angle).

Example. Show that

sec²(θ) = tan²(θ) + 1

Answer. By definitions of the trigonometric functions we have

sec(θ) = and

We have:

Using the magic identity we get

. This completes our proof.

Solutions from the Previous Issue: 

Solve the following equations:

1.       

Solution: 

We use:     and  , also      and    . We have   

 So  .

2.     .

Solution: 

If    (1) and    

(2 ) we have     

. That’s why there are two situations  

a)   and 

b)   but in this situation we must eliminate     see you (1) .  Finally   is the second set of solutions of the initial equation.

3.  .

Solution: 

Let   we have    .  So  .

4.   .

Solution: 

We have , so that . Then  So .

5.    .

Solution: 

  and  . Then we have:   

. In the last proportion we add in the both sides, respectively, denominator to the nominator, and we get an equivalent proportion:

. Dividing both side of the equation by  ,  we pass to an equivalent equation  . Now introducing the designation      we get the quadric equation   whose roots are    and   . Thus the solutions of the initial equation reduces to the solutions of two trigonometric equations:     and    . 

Proposed Problems:

1.       Find the terms a1, a3, a5, a6 for the next arithmetic progression:

                                              a1, -7, a3, -1, a5, a6, 8, ... .

2.        A sequences is given such that  and  ; evaluate .

3.        If the sum of the first and the fifth term of an arithmetic progression is

        equal to 10 and the product of the third term and the fifth term is equal

         45 find the sum of the first twenty term of the progression.

4.        Solve the equation , knowing that its roots are in a geometric progression.