Angles and Identities

Angles and Their Measurement

Angles are measured in degrees or radians. One full revolution equals 360 degrees or 2π radians.
Converting degrees to radians: multiply by π/180.
Converting radians to degrees: multiply by 180/π.
Positive angles are measured anti-clockwise from the positive x-axis, while negative angles are measured clockwise.
Acute angles are less than 90 degrees, right angles are exactly 90 degrees, obtuse angles are between 90 and 180 degrees, and reflex angles are between 180 and 360 degrees.

There are three basic trigonometric functions: sine (sin), cosine (cos), and tangent (tan).
They can be defined for any angle in the unit circle as: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = sin θ / cos θ.
The reciprocal functions are cosecant (csc) which is 1/sin, secant (sec) which is 1/cos, and cotangent (cot) which is 1/tan.

These are derived from the Pythagorean theorem and they include: sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, and 1 + cot² θ = csc² θ.

These are used when dealing with expressions that have a double angle, e.g., sin 2θ, cos 2θ.
They are defined as:
- Sin 2θ = 2 sin θ cos θ
- Cos 2θ = cos² θ - sin² θ = 2 cos² θ - 1 = 1 - 2 sin² θ
Double angle identities can be used to simplify trigonometric equations and to reduce the power of sin and cos in double angle expressions.

These are derived from the complementary nature of the sin and cos functions. They give you quick ways to switch between sin and cos functions in an equation.
They are defined as: sin (90 - θ) = cos θ and cos (90 - θ) = sin θ.