You
should be more afraid of a stupid man than of an evil one.
 Christina of Sweden
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:
Proposed by
Diana Rosu – student St. Thomas Aquinas Secondary School, Brampton, Ontario
1
Solution:
We substitute
and we get
which is a quadric equation
, which roots are
and
. Therefore
and
.
2
Evaluate
Solution:
3
Solve the exponential equation:
Solution:
The equation becomes
. We designate
and we get the equation
whose roots are
,
and
.
4
Solve the system
Solution:
First
and we have
. Thus
and substituting in the first
equation of the system we get
and
.
Proposed exercises:
Solve for x in the following
equation:
1.
12e^{2x } 37e^{x}
+ 21 = 0
2.
e^{2x } 3e^{x}
+ 2 = 0
3.
e^{x }+ 5 = 60
4.
e^{x } 15 = 60
