The Unit Circle and Graphing Section 2.1

The unit Circle When we defined the 6 functions we used any point(x,y) on the terminal side of the angle and we used the distance formula to find r. With the unit circle we will use (x,y) on the unit circle where the radius r is 1. So if 𝛼 is an angle in standard position whose terminal side intersects the unit circle at (x,y) then, sin 𝛼= y cos 𝛼= x tan 𝛼= 𝑦 𝑥 csc 𝛼= 1 𝑦 sec 𝛼= 1 𝑥 Sin 𝛼= 𝑥 𝑦 Provided no denominator is zero.

Evaluating trigonometric functions with the unit circle We can use the mage to the left to evaluate the common angles. Sin 45° Cos( 2𝜋 3 ) Tan( 𝜋 2 ) Sec( 𝜋 3 )

Graphing the simplest Sine Functions Sketch the graph of y=sin x on the interval [0,2𝜋] Make a table of values (select five key values) x O 𝝅 𝟐 𝝅 𝟑𝝅 𝟐 𝟐𝝅 Sin x 1 -1

Example Sketch the graph of y=4sin x on the interval [0,2𝜋]

Period

Graphing a sine function Graph y=2sin x for x in the interval [-2𝜋, 2𝜋] and determine the range. x O 𝝅 𝟐 𝝅 𝟑𝝅 𝟐 𝟐𝝅 Sin x 1 -1 2 sin x 2 -2

Amplitude Amplitude – is the absolute value of half the difference between the minimum and maximum y values Find the amplitude for the function y = 2sin x Find the amplitude for the function y = 8 sin x

Graphing the simplest cosine function Graph y = cos x x O 𝝅 𝟐 𝝅 𝟑𝝅 𝟐 𝟐𝝅 Cos x 1 -1

Transformations of the graph The amplitude of y=A sin x or y= A cos x is 𝐴 Phase shift of the graph of y=sin (x-C) or y=sin (x-C) is C

Example Name the amplitude, phase shift and range of the graph.

Example Name the amplitude, phase shift and range of the graph.

Horizontal shift For y=sin (x) + D or y=cos (x) + D where D<0 is downward and D>0 is upward Graph of y=cos(x-pi/4)+2

Questions