Mathematics Magazine for Grades 1-12  

1/2004

You should be more afraid of a stupid man than of an evil one.
 - Christina of Sweden

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: 

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.        12e2x - 37ex + 21 = 0

2.        e2x - 3ex + 2 = 0

3.        ex + 5 = 60

4.        ex - 15 = 60