MHF 4UI UNIT 6
Trigonometric Functions II
Day 1 ­ Trig Identities (Part I)
Function, Sine
Review: Related Angles:
a θ
θ
a
θ = π + a
θ = π - a
Using CAST Rule :
sin (π-a) = sin a
cos(π-a) = - cos a
tan (π-a) = - tan a
sin (π+a) =- sin a
cos(π+a) = - cos a
tan (π+a) = tan a
θ
a
θa
θ = 2π - a θ = - a
sin (2π-a) = - sin a
cos(2π-a) = cos a
tan (2π-a) = - tan a
sin (-a) = - sin a
cos(-a) = cos a
tan (-a) = - tan a
Example 1: Find the exact value of the following:
a) b)
Example 2: If and ,find θ.
Co ­ Related Angles
(These angles are related to the y axis!!!)
x
y
z
a
θ
x
y
Co ­ Related Angles
a
θ
a
θ
θ
a
θ = π/2 - a
θ = π/2+a
θ = 3π/2 - a
θ
a
θ =3π/2 + a
sin (3π/2+a) = -cos a
sin (π/2+a) = cos a
cos(3π/2+a)
= sin a
cos(π/2+a) = -sin a
tan (3π/2+a) = -cot a
tan (π/2+a) = -cot a
sin (π/2-a) = cos a
sin (3π/2-a) = -cos a
cos(π/2-a) = sin a
cos(3π/2-a) = -sin a
tan (π/2-a) = cot a
tan (3π/2-a) = cot a
Example 3: Express as a co­related angle, then evaluate.
Review: Trig Ratios Using x, y and r
P (x, y)
θ
x2 + y2 = r2
sin θ =
cos θ =
tan θ =
Pythagorean Identity: cos2θ + sin2θ = 1
Example 4: Prove 1 + tan2x = sec2x
Example 5: Prove 1 + cot2x = csc2x