I can calculate trigonometric functions on the Unit Circle

Six Trig Functions We will now define the 6 trig functions for ANY angle. (Not just positive acute angles.) Let θ 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 θ = x/r sinesinsin θ = y/r tangenttantan θ = y/x x≠0 secantsecsec θ = r/x x≠0 cosecantcsccsc θ = r/y y≠0 cotangent cotcot θ = x/y y≠0

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

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

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 θ = x sinesinsin θ = y tangenttantan θ = y/x x≠0 secantsecsec θ = 1/x x≠0 cosecantcsccsc θ = 1/y y≠0 cotangent cotcot θ = x/y y≠0

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

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

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

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

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 (11π/6) csc (-7π/4)

Given that tan Ө = -3/4 and cosӨ > 0, find sinӨ and secӨ. Given that π/2 < Ө < π and that sin Ө = 1/3, find cosӨ and tanӨ.

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?

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

Let θ = 30°, find sec θ 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 don’t these work ?

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