Trigonometric Functions - Contents Radian Measure Area/Circumference/Length of Arc Area of a Sector Area of a minor segment Small Angles Trigonometric Results Trigonometric Graphs Graphical Solutions Derivatives of Trigonometric Functions Integrals of Trigonometric Functions (Right Click Mouse  Pointer Options  Arrow Options  Visible)

Radian Measure 1 Unit 1 Unit 1c One radian is the angle that a one unit arc makes with centre of a unit circle. Proof Circumference = 2πr 2π = 360o π = 180o = 2π (1) π radians = 180o = 2π  The circumference is 2π

Radian Measure Conversions Example 1 Convert into degrees. 5π 3 Example 2 Convert 45o into radians. 5π 3 = 5x180 3 π = 180o = 300o 180o = π radians 1o = π/180 radians Example 3 Convert 1.6c into degrees. 45o = π/180 x 45 radians = 45π/180 radians π radians = 180o = π/4 radians 1 radians = 180/π deg 1.6 radians = 180/π x 1.6 deg = 91o 40’

Area/Circumference/Length of Arc Circumference = 2r = D  Area = r2 r Length of Arc l = r  ( is in radians)

Area/Circumference/Length of Arc Circumference = 2r = D = 2 x 3 = 6 (exact)  4 ≈ 18.8 cm (Approximate) Area = r2 3 cm =  x 32 = 9 (exact) Length of Arc ≈ 28.3 cm2 (Approximate) l = r  ( is in radians) = 3 x /4 = 3/4 (exact) ≈ 2.4 cm (Approximate)

Area of a Sector Area of Sector A = ½ r2  Proof Example ( is in radians) Area of Sector Area of Circle = Angle  Revolution Example Find the area of a sector with radius 3cm and angle /6. A r2 =  2 A =  r2 2 A = ½ r2  A = ½ r2  = ½ 32 x /6. = 3/4 cm2. ≈ 2. 36 cm2.

Area of a Minor Segment Area of Sector A = ½ r2 ( - sin ) Example b a Area = ½ab sin  A = ½ r2 ( - sin ) ( is in radians) Example Find the area of a minor segment formed with radius 3cm and angle /6. Proof Area of Minor Segment = Area of Sector - Triangle = ½ r2  - ½ r2 sin  = ½ r2 ( - sin ) = ½ r2 ( - sin ) = ½ 32 x (/6 - sin /6) = ½ 32 x (/6 - ½) = (3/4 – 3/8)cm2 ≈ 1. 98 cm2.

What happens to a/h as x 0. Small Radian Angles For small angles Sin x ≈ x h Tan x ≈ x x o a=1 Cos x ≈ 1 Therefore Cos x = a/h lim Sin x = 1 x x0 What happens to a/h as x 0. lim Tan x = 1 x x0 a/h  1/1 1

What happens to o/h as x 0. Small Radian Angles For small angles Sin x ≈ x h Tan x ≈ x x o a=1 Cos x ≈ 1 Therefore Sin x = o/h lim Sin x = 1 x x0 What happens to o/h as x 0. lim Tan x = 1 x x0 o/h x  0/1  0

What happens to o/a as x 0. Small Radian Angles For small angles Sin x ≈ x h Tan x ≈ x x o a=1 Cos x ≈ 1 Therefore Tan x = o/a lim Sin x = 1 x x0 What happens to o/a as x 0. lim Tan x = 1 x x0 o/a  0/1  0

Small Radian Angles Example: Evaluate: lim Sin 5x x lim Sin 5x = lim 5(Sin 5x) 5x x0 5 lim (Sin 5x) 5x x0 = = 5 x 1 = 5

Trigonometric Results π/2 π - θ θ 2nd 1st S A Note: π=180o π 2π T C 3rd 4th π + θ 2π - θ 3π/2

Trigonometric Results Sin π/3 = √3/2 Sin π/6 = 1/2 Cos π/3 = 1/2 Cos π/6 = √3/2 60o π/3 Tan π/3 = √3 Tan π/6 = 1/√3 π/6 2 2 √3 60o π/3 60o π/3 1 2 π/3 = 60o π/6 = 30o

Trigonometric Results Sin π/4 = 1/√2 Cos π/4 = 1/√2 π/4 45o Tan π/4 = 1 √2 1 π/4 45o 1 π/4 = 45o

Trigonometric Results A S C T Sin is -ve in the 3rd Quadrant. Find the exact value of: 5π 4 5x180o 4 Sin Sin = = Sin 225o = Sin (180o + 45o) = -Sin 45o = -1 √2

Trigonometric Results Solve for 0 ≤ θ ≤ 2π A S C T Evaluate Sin θ = 1/2 Sin is +ve in the 1st & 2nd Quadrants. θ = 30o 30o 2 1 θ = π/6 θ = π - π/6 θ = 5π/6

Trigonometric Graphs y = sin(x) π/2 3π/2 Geogebra

Trigonometric Graphs y = cos(x) π/2 3π/2 Geogebra

Trigonometric Graphs y = tan(x) π/2 3π/2 Geogebra

Trigonometric Graphs y = cosec(x) π/2 3π/2 Geogebra

Trigonometric Graphs y = sec(x) π/2 3π/2 Geogebra

Trigonometric Graphs y = cot(x) π/2 3π/2 Geogebra

Trigonometric Graphs The a Sin x Family of Curves y = sin(x) Geogebra

Trigonometric Graphs The Sin bx Family of Curves y = sin 3x y = sin 2x Geogebra

Trigonometric Graphs The Sin x + c Family of Curves y = sin x Geogebra

Graphical Solution Graphical solve y = cos x and y = x y = x y = cos x One Solution π/4 π/2

Derivative of Trigonometric Functions Function of Function Rule Derivative Sin x d dx [sin f(x)] = f’(x) cos f(x) d dx (sin x) = cos x Cos x d dx (cos x) = sin x d dx [cos f(x)] = -f’(x) sin f(x) Tan x d dx (tan x) = sec2 x d dx [tan f(x)] = f’(x) sec2 f(x)

Derivative of Trigonometric Functions Derivative Examples Sin 4x d dx (sin 4x) = 4 cos 4x d dx [sin f(x)] = f’(x) cos f(x) Tan (3x + 1) d dx [tan (3x + 1)] = 3 sec2 (3x + 1) d dx [tan f(x)] = f’(x) sec2 f(x) Cos (x2 - 2x + 1) d dx [cos (x2 - 2x + 1)] = -(2x - 2) sin (x2 -2x + 1) = 2(1 - x) sin (x2 -2x + 1) d dx [cos f(x)] = -f’(x) sin f(x)

Integrals of Trigonometric Functions Function of Function Rule Integral Sin x sin x dx = - cos x + C ∫ ∫ sin (ax +b) dx = - 1/a cos (ax + b) + C Cos x cos x dx = sin x + C ∫ cos (ax +b) dx = 1/a sin (ax + b) + C ∫ Tan x sec2 x dx = tan x + c ∫ sec2 (ax +b) dx = 1/a tan (ax + b) + C ∫

Integral of Trigonometric Functions Integration Examples ∫ sin (ax +b) dx = - 1/a cos (ax + b) + C ∫ sin (3x +2) dx = -1/3 cos (3x + 2) + C

Integral of Trigonometric Functions Integration Examples sin x dx = - cos x + C ∫ ∫ sin x dx = π 2 - cos x π 2 = - cos - (-cos 0) π 2 = 0 – (-1) = 1