Slide 1

I can calculate trigonometric functions on the Unit Circle

Slide 2

Six Trig Functions We will now define the 6 trig functions for ANY angle. (Not just positive acute angles.) Let be any angle in standard position and (x,y) a point on its terminal ray. Let the distance from the point to the origin be r. Then Name AbbreviationDefinition cosinecoscos = x/r sinesinsin = y/r tangenttantan = y/x x0 secantsecsec = r/x x0 cosecantcsccsc = r/y y0 cotangent cotcot = x/y y0

Slide 3

Find the value of the six trig functions for the angle whose terminal ray passes through: a)(-5,12) b) (-3,-3)

Slide 4

The Unit Circle The unit circle is given by x 2 + y 2 = 1. Graph this. Now choose an angle which intersects the unit circle at (x,y). Draw it.

Slide 5

We will now define the 6 trig functions for any NUMBER t on a number line. Let (x,y) be the point on t when the number line is wrapped around the unit circle. (What happened to r?) Name AbbreviationDefinition cosinecoscos = x sinesinsin = y tangenttantan = y/x x0 secantsecsec = 1/x x0 cosecantcsccsc = 1/y y0 cotangent cotcot = x/y y0

Slide 6

Where is the triangle? Compute the six trigonometric functions for = . cos = -1 sin = 0 tan = 0 sec = -1 csc is undefined cot is undefined Check your answers on your calculator. Make sure you are in the correct mode.

Slide 7

A few more Find the tan 450 Find the cos of 7/2

Slide 8

Can you make a chart showing in which quadrants sin, cos, and tan have positive values and in which quadrants they have negative values?

Slide 9

Reference Angles For an angle in standard position, the reference angle is the acute, positive angle formed by the x- axis and the terminal side of . Give the reference angles for 135 5/3 210 /4.

Slide 10

An angle will share the same x and y coordinates with its reference angle, but the signs may be different. (Can you see why?) Find cos 150 tan 135 cot (-120) cos (11/6) csc (-7/4)

Slide 11

Given that tan = -3/4 and cos > 0, find sin and sec. Given that /2 < < and that sin = 1/3, find cos and tan.

Slide 12

To think about Can you explain why sine and cosine must be between -1 and 1? Must the other functions also lie in that interval?

Slide 13

Which is greater sin 2 or sin 2? Which is greater cos 2 or cos 2?

Slide 14

Let = 30, find sec using your calculator. Answer: 1.15 Make sure you are in degree mode. Then either 1/ (cos 30) or (cos 30) -1. NOT cos (30 -1 ) or cos -1 (30). Why dont these work ?

Slide 15

TRUE or FALSE? cos (- ) = cos( ) sin (- ) = -sin( ) Both statements are true. Justify with a picture.