# Table of Contents

# Introduction

Most students are familiar with the drawing in Figure 1 used
to define sin*θ* and cos*θ* for an angle
*θ*.

Figure 1. The sine and cosine of a given angle *θ*.

From this diagram, we may deduce, using the Pythagorean theorem, that
sin^{2}*θ* + cos^{2}*θ* = 1.

Unfortunately, at this point, high-school teachers will then proceed
to simply define the other trigonometric functions using algebra formulae
(e.g., tan(*θ*) = sin(*θ*)/cos(*θ*) )
without any reference to the geometry.

# Tangent (tan) and Secant (sec)

To begin, in Figure 2 we draw a line (red) which is tangent to the point (1, 0) (the point labeled 1)
and we extend the line __0a__ to intersect this tangent line
at b. The length of the line __1b__ is by definition tan*θ*
and the length of the line __0b__ is sec*θ*.

Figure 2. The tangent and secant of a given angle *θ*.

With these definitions, we may use similar triangles and the Pythagorean
theorem to come up with a full set of identities:

- tan
^{2}*θ* + 1 = sec^{2}*θ*.
- tan
*θ*:1 = sin*θ*:cos*θ* ⇒ tan*θ* = sin*θ*/cos*θ*.
- sec
*θ*:1 = 1:cos*θ* ⇒ sec*θ* = 1/cos*θ*.
- sec
*θ*:tan*θ* = 1:sin*θ* ⇒ sin*θ* = tan*θ*/sec*θ*.
- (tan
*θ* - sin*θ*)^{2} + (1 - sin*θ*)^{2} = (sec*θ* - 1)^{2}.

To figure out where the last identity came from, look for a small right-angled
triangle.

# Complementary Tangent (cot) and Complementary Secant (csc)

Similarly, we can draw a line (in blue) which is tangent to the point
(0,1) (the point labeled 1'), as shown in Figure 3 and extend the
line __0a__ until it intersects the complementary tangent line
at c. The length of the line __1'c__ is by definition the complementary
tangent, or cotangent of *θ* and the length of the line __0c__ is
the complementary secant of *θ*, or sec*θ*.

Figure 3. The cotangent and cosecant of a given angle *θ*.

Yet again, we may derive all the various identities:

- cot
^{2}*θ* + 1 = csc^{2}*θ*.
- cot
*θ*:1 = cos*θ*:sin*θ* ⇒ cot*θ* = cos*θ*/sin*θ*.
- csc
*θ*:1 = 1:sin*θ* ⇒ csc*θ* = 1/sin*θ*.
- csc
*θ*:cot*θ* = 1:cos*θ* ⇒ cos*θ* = cot*θ*/csc*θ*.
- (cot
*θ* - cos*θ*)^{2} + (1 - cos*θ*)^{2} = (csc*θ* - 1)^{2}.

In addition to these identities, we may also come up with the following identities
between the various functions:

- cot
*θ*:1 = 1:tan*θ* ⇒ cot*θ* = 1/tan*θ*.
- csc
*θ*:1 = sec*θ*:tan*θ* ⇒ csc*θ* = sec*θ*/tan*θ*.
- csc
*θ*:cot*θ* = sec*θ*:1 ⇒ sec*θ* = csc*θ*/cot*θ*.

# ½π - θ Formula

If you take a look at Figure 4, you may also observe that the line
which is the *tangent* for *θ*
is the cotangent line for *π*/2 - *θ*, and therefore
we see that the definitions for tangent and secant for θ are
equal to the definitions of cotangent and cosecant for ½π - θ,
and vice versa.

Figure 4. Relationship between *θ* and *π*/2 - *θ*.

From this comes the relationships:

- sin(
*π*/2 - *θ*) = cos*θ*.
- cos(
*π*/2 - *θ*) = sin*θ*.
- tan(
*π*/2 - *θ*) = cot*θ*.
- sec(
*π*/2 - *θ*) = csc*θ*.
- csc(
*π*/2 - *θ*) = sec*θ*.
- cot(
*π*/2 - *θ*) = tan*θ*.

# Approximations

Additionally, we can see why for small values of *θ* (in radians):

- sin
*θ* ≈ *θ*,
- cos
*θ* ≈ 1,
- tan
*θ* ≈ *θ*,
- sec
*θ* ≈ 1,
- csc
*θ* ≈ ∞, and
- cot
*θ* ≈ ∞.

Recall that the value *θ* in radians is the length of the
curve __1a__ shown in Figure 1.