# Section
2.8 Properties of Trigonometric Functions

## (Unfinished -- in progress)

- Circles: Arc length and sector area

- Argument vs. independent variable

- Periodicity with respect to argument vs. wrt independent variable.

- sin(π/2 - θ)= cos(θ), cos(π/2 - θ) = sin(θ), tan(π/2 - θ) = 1/tan(θ)

- sin(π - θ) = sin(θ), cos(π - θ) = -cos(θ), tan(π - θ) = -tan(θ)

- addition formulas for sin, cos, and tan and implications of them

## Exercise Set 2.8

*Figure 2.8.1. Relationships among
sine, cosine, and angles that differ by ** $\frac{\pi}{2}$*.

1. Examine the diagrams in Figure 2.8.1. Use the Angle-Side-Angle theorem
from geometry to prove that the two triangles are congruent, and therefore
that $\cos(θ) = \sin(θ + \frac{\pi}{2})$ and that
$\sin(θ) = -\cos(θ+\frac{\pi}{2})$.

2. Now that you have proved the statements in Exercise 2.7.1, use the
animation in Figure 2.8.2 to convince yourself that they are always true.
Pause the movie occasionally and examine the relationships shown in Figure
2.8.1. In what way does the diagram give compelling visual confirmation that
$\cos(θ) = \sin(θ + \frac{\pi}{2})$ and that
$\sin(θ) = -\cos(θ+\frac{\pi}{2})$ for all values of θ?

*Figure 2.8.2. Relationships among sine, cosine, and angles that differ
by $\frac{\pi}{2}$.*

3. The functions secant, cosecant, and cotangent are defined as
$\sec(x) = \frac{1}{\cos(x)}$, $\csc(x) = \frac{1}{\sin(x)}$, and
$\cot(x) = \frac{\cos(x)}{\sin(x)}$. Use the fact that $\cos(x) =
\sin(x + \frac{\pi}{2})$ to express each of sec(*x*), csc(*x*),
cot(*x*), and tan(*x*) completely in terms of sin(*x*). This
demonstrates that sin(*x*) is the only elementary trigonometric
function we need, and that all other trigonometric functions can be defined
in terms of it.

4. In the unit circle, $(\sin x)^2 + (\cos x)^2 = 1$ by the
Pythagorean Theorem.

a) Use this fact to derive an equivalent
identity involving tan(*x*).

b) Use this fact to derive an equivalent identity involving cot(*x*).

c) Use this fact to derive an equivalent identity involving csc(*x*).

d) Use this fact to derive an equivalent identity involving sec(*x*).