Mathematics Magazine for Grades 112 



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) cos^{2}(θ)
+ sin^{2}(θ) = 1 where θ
is any real number (of course θ measures an angle). Example.
Show that sec^{2}(θ)
= tan^{2}(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. 


