5.2-The Unit Circle & Trigonometry

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The Unit Circle 45 o 225 o 135 o 315 o 30 o 150 o 110 o 330 o π6π6 11π 6 5π65π6 7π67π6 7π47π4 π4π4 5π45π4 3π43π4 π3π3 2π32π3 4π34π3 5π35π3 0 π2π2 π 3π23π2 120 o 180 o 90 o 60 o 0o0o 240 o 270 o 300 o

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5.2—How the unit circle relates to Trigonometry Trigonometric Functions NameAbbreviations Sine sin Cotangent cot Cosine cos Secant sec Cosecant csc Tangent tan

The circular function definitions of the six trigonometric functions: *Let t be any real number, and (x,y) the point on the unit circle corresponding to t:

Examples: Evaluate all six trig. functions for each value of t below: 1.

The Unit Circle 45 o 225 o 135 o 315 o 30 o 150 o 110 o 330 o π6π6 11π 6 5π65π6 7π67π6 7π47π4 π4π4 5π45π4 3π43π4 π3π3 2π32π3 4π34π3 5π35π3 0 π2π2 π 3π23π2 120 o 180 o 90 o 60 o 0o0o 240 o 270 o 300 o

Examples: Evaluate all six trig. functions for each value of t below: 2.

What is the a.) What is the domain of sin t? Of cos t? b.) maximum value of sin t? Of cos t? c.) minimum value of sin t? Of cos t? d.) What is the range for sin t and cos t?

The Unit Circle 45 o 225 o 135 o 315 o 30 o 150 o 110 o 330 o π6π6 11π 6 5π65π6 7π67π6 7π47π4 π4π4 5π45π4 3π43π4 π3π3 2π32π3 4π34π3 5π35π3 0 π2π2 π 3π23π2 120 o 180 o 90 o 60 o 0o0o 240 o 270 o 300 o

Domain: Range: * Recall this notation: brackets [ ] means end points are INCLUDED

Examples: Evaluate all six trig. functions for each value of t below: 3.

even function: odd function: From the last example which trigonometric functions would you guess are even? Which would you guess are odd?

Ex 1.) 2 Ex 3.) -2 Let be t and write - as - t

The cosine and secant functions are Even. The sine, cosecant, tangent and cotangent functions are Odd.

Examples: 1. Given and find each value: sin (-t) = csc (t) = csc (-t) = sin (t + π) = sin (π – t) =

Examples: 1. Given and 1 (x, ¾ ) t -t (x, -¾ ) = - ¾ 0 π2π2 π 3π23π2

Examples: 1. Given and find each value: = - ¾

Examples: 1. Given and 1 (x, ¾ ) t (-x, -¾ ) sin (t + π) = -¾ t+π

Examples: 1. Given and (x, ¾ ) -t (-x, ¾ ) sin (π-t) = ¾ π π-t sin t = ¾

Examples: 1. Given and 1 (x, ¾ ) t (-x, -¾ ) sin (t + π) = -¾ t-π sin (π-t) = sin (-(-π+t) ) = sin -(t-π) = -sin (t-π) = -( -¾) = ¾

2. Given and find each value: cos t = sec t = cos (π + t) = cos (t – π) =

Homework: p. 361:(2-16) Evens, 19, 21 (31-36) all Quiz 5.1 – 5.2 Friday: (I’ll give you a blank unit circle to fill in.) Before you leave finish #16, 21, 31, 35 and show these problems to me so that I Can check before you go.