12-3 Trigonometric Functions of General Angles Part 1: Points and Reference Angles

Trig Functions for Any Angle Previously we have defined trig functions for acute angles only; now we will use angles of any measure (in standard position). For any angle θ and any point (x, y) on the terminal side and let r be the distance from the origin to (x, y): y y y (x, y) r θ θ θ x x x r (x, y) r (x, y)

Finding Trig Functions Using a Point Find the values of the six trig functions of an angle θ in standard position whose terminal side passes through (8, -15).

When an angle θ is in standard position its terminal side passes through (-3, -4). Find the values of the six trig functions of θ.

Signs of Functions In the previous definitions, r will always be positive, so the sign of the function will be determined by the sign of x or y, which depend on the quadrant of the terminal side of θ. *sine and cosine are defined for all angles, but the other four are undefined for certain quadrantal angles Function value Quadrant of θ 1 2 3 4 sin θ csc θ + - cos θ sec θ tan θ cot θ

Determining Defined Functions Determine which functions are defined for a 180 angle and find their values.

Determine which functions are defined for a 90 angle and find their values.

Reference Angles If θ is not a quadrantal angle, there is a unique acute angle α, corresponding to θ, formed by the terminal side of θ and the x-axis (positive or negative, depending on the location of the terminal side). When θ is in standard position, α is called the reference angle of θ. θ θ α α α α θ θ θ = -210 α = 30 θ = 420 α = 60 θ = 300 α = 60 θ = 225 α = 45

Finding Reference Angles Find the measure of the reference angle α for each given angle θ. θ = 140 θ = 30010’ θ = -135 How would you find α for a positive angle in quadrant I? Negative?

For each angle θ, find its reference angle. θ = 170 θ = 250 θ = -305

Functions of Acute Angles Write cos 200 as a function of an acute angle.

Write sin 130 as a function of an acute angle.