A Review of Trigonometric Functions

Right Triangle Vocabulary hypotenuse c opposite adjacent a A C adjacent opposite b

Trigonometric Functions Defined in terms of right triangles sin(x) = opp/hyp cos(x) = adj/hyp tan(x) = opp/adj = sin(x) / cos(x) Know the graphs

Trigonometric Functions Defined in terms of the unit circle P(x)=(cos x, sin x) 1 sin x x cos x 1

Other Trig Functions cot(x) = 1/tan(x) = cos(x) / sin(x) sec(x) = 1/cos(x) csc(x) = 1/sin(x)

Odd/Even Odd Sin(x) Csc(x) Tan(x) Cot(x) Even Cos(x) Sec(x)

Radians Radian measure of the angle at the center of a unit circle equals the length of the arc that the angle cuts from the unit circle.  1 C

Radians s r =  1 s s r =   1 r C Note: Radian measure is a dimensionless number

Radians and Degrees s 2 r =  2 radian measure 2 arclength circumference degree measure 360° = = 2 = 360°  = 180°

Famous Values Angle 0º = 0 30º = /6 45º = /4 60º = /3 90º = /2 Sin 1 2 3 4 /2 =0 =1/2 =1 Cos 4 3 2 1 /2 =1 =1/2 =0 Tan 1/ 3 1 3 Und

Domain, Range, Period sin(x) cos(x) tan(x) (-, ) x /2, 3/2, ... (-1, 1) (-, ) Period 2 

Finding the Period Sin (3πx/2 + 4) Set term that includes x equal to the period of the trig function 3πx/2 = 2π Solve for x x = 4/3 = Period

Trig Identities to Know Pythagorean Identities sin2 x + cos2 x = 1 tan2 x + 1 = sec2 x cot2 x + 1 = csc2 x Double Angle sin(2x)=2sin(x)cos(x) cos(2x)=cos2(x) – sin2(x) Square sin2(x) = (1 – cos(2x))/2 cos2(x) = (1 + cos(2x))/2

Creating Inverse Trig Functions The trig functions are not 1-1 Restrict their domains y = sin(x) -π/2 ≤ x ≤ π/2 y = cos(x) 0 ≤ x ≤ π y = tan(x) -π/2 < x < π/2

The Inverse Trig Functions y = sin-1 x or y = arcsin(x) Domain: [-1, 1] Range: [-π/2, π/2] y = cos-1 x or y = arccos(x) Range: [0, π] y = tan-1 x or y = arctan(x) Domain: (-∞, ∞) Range: (-π/2, π/2)

Examples sin x = 0.455 x = sin-1 (0.455) cos x = π/2 x = cos-1 (π/2) tan x = 8 x = tan-1 (8)