# TRIGONOMETRY

### >Right triangle trigonometry

A *right triangle* is defined as having one angle precisely equal to 90^{o} (a *right angle*).

### Trigonometric identities

H is the *Hypotenuse*, always being opposite the right angle. Relative to angle x, O is the *Opposite* and A is the *Adjacent*.

“Arc” functions such as “arcsin”, “arccos”, and “arctan” are the complements of normal trigonometric functions. These functions return an angle for a ratio input. For example, if the tangent of 45^{o} is equal to 1, then the “arctangent” (arctan) of 1 is 45^{o}. “Arc” functions are useful for finding angles in a right triangle if the side lengths are known.

### The Pythagorean theorem

### Non-right triangle trigonometry

### The Law of Sines (for *any* triangle)

### The Law of Cosines

## Trigonometric equivalencies

## Hyperbolic functions

Note: all angles (x) must be expressed in units of *radians* for these hyperbolic functions. There are 2π radians in a circle (360^{o}).

Solution:Step1First replace x with (x-Δx) in the given fictnuon f(x) to obtainf(x+Δx) = 2(x+Δx)+1 = 2x + 2Δx + 1 Step2Rewrite f(x-Δx) as (2x + 2Δx + 1) and f(x) as (2x + 1) into the definition of the derivative limit to obtain: lim [(2x + 2Δx + 1) – (2x+1)] / ΔxΔx-> 0 Step 3Combine like terms on the numerator and the reduced expression becomes lim [2Δx] / ΔxΔx-> 0 Step 4Cancel out Δx from both numerator and denominator and finally you get the derivative Derivative of f (x) = 2x+1 = 2 -Note there was no need to substitute Δx with 0 since it was canceled out.

Solution:Step1First replace x with (x-Δx) in the given fictnuon f(x) to obtainf(x+Δx) = 2(x+Δx)+1 = 2x + 2Δx + 1 Step2Rewrite f(x-Δx) as (2x + 2Δx + 1) and f(x) as (2x + 1) into the definition of the derivative limit to obtain: lim [(2x + 2Δx + 1) – (2x+1)] / ΔxΔx-> 0 Step 3Combine like terms on the numerator and the reduced expression becomes lim [2Δx] / ΔxΔx-> 0 Step 4Cancel out Δx from both numerator and denominator and finally you get the derivative Derivative of f (x) = 2x+1 = 2 -Note there was no need to substitute Δx with 0 since it was canceled out.