Graphing Trig Functions Trigonometry Graphing Trig Functions Javascript

Graphs of the Trig Functions y = sin x Fluctuates from 0 to a high of 1, down to –1, and back to 0, in a space of 2. 1 –2π –π π 2π –1

90° 120° 60° 135° 45° 150° H 30° O O H 180° 0° A A 210° 330° 225° 315° 240° 300° 270°

315° 225° 270° 0° 180° 90° 45° 30° 330° 300° 60° 240° 135° 210° 150° 120° y = sin x 1 90° 180° 270° 360° –1

315° 225° 270° 0° 180° 90° 45° 30° 330° 300° 60° 240° 135° 210° 150° 120° y = sin x 90° 180° 270° 360°

Graphs of the Trig Functions Cosine The graph of y = cos x resembles the graph of y = sin x but is shifted, or translated, units to the left. It fluctuates from 1 to 0, down to –1, back to 0 and up to 1, in a space of 2. 1 –2π –π π 2π –1

y = tan x 270° 0° 180° 90° – 90° 0° 90° 180° 270° 360° 450° 315° 225° 45° 30° 330° 300° 60° 240° 135° 210° 150° 120° y = tan x – 90° 0° 90° 180° 270° 360° 450°

Inverse Trig Functions Trigonometry Inverse Trig Functions Definition:  For all one-to-one functions, the inverse function is the set of ordered pairs obtained by interchanging the first and second elements of each pair in the original function. Notation:   If  f  is a given function, then f -1 denotes the inverse of  f.

Inverse Trig Functions

Graph of y = x 2 -2 2 10 y 8 6 4 x f (2) = 4 and f (–2) = 4 so what is an inverse function supposed to do with 4? By definition, a function cannot generate two different outputs for the same input, so this function does not have an inverse. Horizontal line test

Graph of y = x 2 f (x) = x2 for x ≥ 0 f –1(x) = x y -2 2 10 y 8 6 4 x By taking only one half of the graph: x ≥ 0 , the graph now passes the horizontal line test and we do have an inverse. f (x) = x2 for x ≥ 0 f –1(x) = x Note how each graph reflects across the line y = x onto its inverse.

y = sin x The sine function does not pass the horizontal line test To overcome this, the value for θ is restricted to – 90º ≤ θ ≤ 90º These are known as the “Principal Values” 1 –2π –π π 2π –1

-1 1 x y π 2 _ __ __ y = sin x

y π 2 __ 1 x -1 1 π 2 _ __ π 2 __ -1 π 2 _ __ y = sin–1 x

The thing to remember is that for the trig function the input is the angle and the output is the ratio, but for the inverse trig function the input is the ratio and the output is the angle.

y = cos x The restriction used for y = sin x in not suitable Domain [0, π] and the range is [–1, 1] 1 –2π –π π 2π –1

y = cos–1 x y p 5p/6 2p/3 p/2 p/3 p/6 x -1 1

y = tan–1 x Like the sine function, the domain of the section of the tangent that generates the inverse tan is -p /2 -p /4 p /4 p /2 -4 -3 -2 1 2 3 4 x y -1 y = tan x y = tan–1x y -4 –2 2 4 6.0 –p /2 –p /4 p /4 p /2

The table below will summarize the parameters we have so far The table below will summarize the parameters we have so far. Remember, the angle is the input for a trig function and the ratio is the output. For the inverse trig functions the ratio is the input and the angle is the output. sin–1x cos–1x tan–1x Domain Range When x < 0, y = sin–1x will be in 4th quadrant When x < 0, y = cos–1x will be in 2nd quadrant When x < 0, y = tan–1x will be in 4th quadrant

The graphs give you the big picture concerning the behavior of the inverse trig functions. Calculators are helpful with calculations. But special triangles can be very helpful with respect to the basics. 1 2 60º 30º 3 1 2 45º

1 2 60º 30º 3 1 2 45º

1 2 60º 30º 3 Negative inputs for the cos–1 can be a little tricky. y x –1 2 1 2 60º 30º 3 Negative inputs for the cos–1 generate angles in the 2nd Quadrant so we have to use 60 degrees as a reference angle in the 2nd Quadrant.

θ θ is an angle whose sine is

y x 1 2

–1 2 14. 1 2 15.

– (i) Copy and complete the table below for f : x → tan–1x, giving the values for f (x) in terms of π. x f (x) –1 1 – 3 3 – (ii) Draw the graph of y = f (x) in the domain –2 ≤ x ≤ 2, scaling the y-axis in terms of π. y -4 –2 2 4 6.0 –p /2 –p /4 p /4 p /2 2006 Paper 2 Q5 (a)