Download Trigonometry

Trigonometry
Trigonometry can seem like hundreds of formulas and identities, but in reality you don’t need to
memorize every single formula. What follows is a reasonable “base-line” knowledge level that
should be adequate for calculus. The more you use it, the better it stays with you (and makes
more sense).
I. Definitions of the Trigonometric Functions.
A circle is drawn with radius 1 and center at the origin. This is called the unit circle. A ray is
drawn from the origin outward. It intersects the circle at point P with angle θ. By definition, the
x-coordinate of this point is the cosine of θ (abbreviated cos θ), and the y-coordinate is the sine
of θ (abbreviated sin θ). The slope of the ray is the tangent of θ (abbreviated tan θ).
There are three other trigonometric functions: the secant of θ (abbreviated sec θ) is the reciprocal
of cos θ; the cotangent of θ (abbreviated cot θ) is the reciprocal of tan θ; and the cosecant of θ
(abbreviated csc θ) is the reciprocal of sin θ.
II. Quadrants and Signs of the Trigonometric Functions.
The xy-plane is split into four quadrants, numbered I, II, III and IV as shown in the figure below.
i) When the ray is in the first quadrant, the x and y coordinates of point P are positive, thus
cos θ and sin θ are positive. The ray also has a positive slope hence tan θ is positive. In
fact, all six trigonometric functions are positive in Q-I.
ii) When the ray is in the second quadrant, the x coordinate of point P is negative and the y
coordinate is positive. Thus, cos θ is negative and sin θ is positive. The slope is negative,
hence tan θ is negative.
iii) When the ray is in the third quadrant, the x and y coordinates of point P are both negative,
so both cos θ and sin θ are negative. The ray has a positive slope so tan θ is positive.
iv) When the ray is in the fourth quadrant, the x coordinate of point P is positive and the y
coordinate negative, so cos θ is positive and sin θ is negative. The slope is negative so tan
θ is negative.
III. Radian and Degree Measure.
The circle is divided into 360 degrees, which is a nice number but has no physically meaningful
interpretation. An alternative measurement called radian measure uses the circumference of the
unit circle so that the angle of the ray that “subtends” the circle is equal to the length of the arc it
creates.
The circumference formula for a circle is C = 2π ⋅r . For the unit circle, the radius is 1, so the
circumference is C = 2π . Since this corresponds to a complete circle (360 degrees), we declare
that
360 degrees = 2π radians
IV. The Trigonometric Ratios
The six trigonometric ratios are defined in the following way based on this right triangle
and the angle θ
V. Special Angles
Here is the proof that in a 30°-60°-90° triangle the sides are in the ratio 1 : 2 :
on the fact that a 30°-60°-90° triangle is half of an equilateral triangle.
. It is based
Draw the equilateral triangle ABC. Then each of its equal angles is 60°.
Draw the straight line AD bisecting the angle at A into
two 30° angles.
Then AD is the perpendicular bisector of BC. Triangle ABD
therefore is a 30°-60°-90° triangle.
Now, since BD is equal to DC, then BD is half of BC.
This implies that BD is also half of AB, because AB is equal to BC. That is,
BD : AB = 1 : 2
From the Pythagorean theorem, we can find the third side AD:
AD2 + 12 = 22
AD2 = 4 − 1 = 3
AD =
.
Therefore in a 30°-60°-90° triangle the sides are in the ratio 1 : 2 :
set out to prove.
; which is what we
Theorem. In an isosceles right triangle the sides are in the ratio 1:1:
.
Proof. In an isosceles right triangle, the equal sides make the right angle. They have the ratio
of equality, 1 : 1.
To find the ratio number of the hypotenuse h, we have, according to the Pythagorean
theorem,
h2 = 12 + 12 = 2.
Therefore,
h=
.
(Therefore the three sides are in the ratio
1:1:
.