convert to radians. leave 345° answer as a multiple of...

A) B)

C) D)

CONVERT TO RADIANS. LEAVE ANSWER

AS A MULTIPLE OF Π.

345°

NONE OF THESE

Thursday, October 27, 2011

A) B)

C) D)

CONVERT TO RADIANS. LEAVE ANSWER

AS A MULTIPLE OF Π.

–60°

NONE OF THESE


A) B)

C) D)

CONVERT RADIANS TO DEGREES.

NONE OF THESE

30° 45°

60°


A) B)

C) D)

CONVERT RADIANS TO DEGREES.

NONE OF THESE

245° 185°

290°


A) B)

C) D)

FIND THE MEASURE, TO THE NEAREST DEGREE, OF THE ACUTE CENTRAL

ANGLE IN THE FIGURE BELOW.

22° 41°

48° 77°


A) B)

C) D)

USE A CALCULATOR TO FIND TAN 4.9382.

–4.3530 –0.2297

0.2239 4.4663


THE UNIT CIRCLE AND CIRCULAR FUNCTIONS

3.3CIRCULAR FUNCTIONS ▪ FINDING VALUES OF CIRCULAR

FUNCTIONS ▪ DETERMINING A NUMBER WITH A GIVEN CIRCULAR FUNCTION VALUE ▪ APPLYING CIRCULAR

FUNCTIONS

MRS. POLANDOCTOBER 26, 2011


CIRCULAR FUNCTIONS

A UNIT CIRCLE HAS ITS CENTER AT THE ORIGIN AND A RADIUS OF 1 UNIT.

THE TRIGONOMETRIC FUNCTIONS OF ANGLE Θ IN RADIANS ARE FOUND BY

CHOOSING A POINT (X, Y) ON THE UNIT CIRCLE CAN BE REWRITTEN AS

FUNCTIONS OF THE ARC LENGTH S.

WHEN INTERPRETED THIS WAY, THEY ARE CALLED CIRCULAR FUNCTIONS.


FOR ANY REAL NUMBER S REPRESENTED BY A DIRECTED ARC ON

THE UNIT CIRCLE,

CIRCULAR FUNCTIONS


THE UNIT CIRCLE

THE UNIT CIRCLE IS SYMMETRIC WITH RESPECT TO THE X-AXIS, THE Y-AXIS, AND THE ORIGIN.

IF A POINT (A, B) LIES ON THE UNIT CIRCLE, SO DO (A,–B), (–A, B) AND (–A, –B).


THE UNIT CIRCLE

FOR A POINT ON THE UNIT CIRCLE, ITS REFERENCE ARC IS THE SHORTEST ARC FROM THE POINT ITSELF TO THE NEAREST POINT ON THE X-AXIS (SAME AS REFERENCE ANGLE, BUT IN RADIANS).

FOR EXAMPLE, THE QUADRANT I REAL NUMBER

IS ASSOCIATED WITH THE POINT ON THE UNIT

CIRCLE.

SINCE SIN S = Y AND COS S = X, WE CAN REPLACE X AND Y IN THE EQUATION OF THE UNIT CIRCLE

TO OBTAIN THE PYTHAGOREAN IDENTITY


CIRCULAR FUNCTION VALUES OF REAL NUMBERS ARE OBTAINED IN THE SAME MANNER

AS TRIGONOMETRIC FUNCTION VALUES OF ANGLES MEASURED IN RADIANS.

THIS APPLIES BOTH TO METHODS OF FINDING EXACT VALUES (SUCH AS REFERENCE ANGLE

ANALYSIS) AND TO CALCULATOR APPROXIMATIONS.

CALCULATORS MUST BE IN RADIAN MODE WHEN FINDING CIRCULAR FUNCTION VALUES.

EVALUATING A CIRCULAR FUNCTION


! EXAMPLE 1 FINDING EXACT CIRCULAR FUNCTION VALUES

EVALUATING A CIRCULAR FUNCTION

AT THE REAL NUMBER IS

EQUIVALENT TO EVALUATING IT AT

RADIANS.

AN ANGLE OF INTERSECTS

THE CIRCLE AT THE POINT (0, –1).

SINCE SIN S = Y, COS S = X, AND

FIND THE EXACT VALUES OF


USE THE FIGURE TO FIND THE EXACT VALUES OF

!EXAMPLE 2(A) ‏ FINDING EXACT CIRCULAR FUNCTION VALUES

THE REAL NUMBER

CORRESPONDS TO THE UNIT

CIRCLE POINT


USE THE FIGURE AND THE DEFINITION OF TANGENT

TO FIND THE EXACT VALUE OF

!EXAMPLE 2(B) ‏ FINDING EXACT CIRCULAR FUNCTION VALUES

NEGATIVE DIRECTION

YIELDS THE SAME ENDING POINT AS MOVING AROUND

THE

MOVING AROUND THE UNIT

CIRCLE UNITS IN THE

CIRCLE UNITS IN THE POSITIVE DIRECTION.


CORRESPONDS TO

!EXAMPLE 2(B) ‏ FINDING EXACT CIRCULAR FUNCTION VALUES


USE REFERENCE ANGLES AND DEGREE/RADIAN

CONVERSION TO FIND THE EXACT VALUE OF

!EXAMPLE 2(C) ‏ FINDING EXACT CIRCULAR FUNCTION VALUES

AN ANGLE OF CORRESPONDS TO AN ANGLE OF 120°.

IN STANDARD POSITION, 120° LIES IN QUADRANT II WITH A REFERENCE ANGLE OF 60°, SO

COSINE IS NEGATIVE IN QUADRANT II.


! EXAMPLE 3 APPROXIMATING CIRCULAR FUNCTION VALUES

FIND A CALCULATOR APPROXIMATION FOR EACH CIRCULAR FUNCTION VALUE.

(A)!COS 1.85 (B)!COS .5149≈ –.2756 ≈ .8703

(C)!COT 1.3209 (D)!SEC –2.9234≈ .2552 ≈ –1.0243


CAUTION A COMMON ERROR IN

TRIGONOMETRY IS USING A CALCULATOR IN DEGREE MODE

WHEN RADIAN MODE SHOULD BE USED.

REMEMBER, IF YOU ARE FINDING A CIRCULAR FUNCTION VALUE OF A REAL NUMBER, THE CALCULATOR

MUST BE IN RADIAN MODE.


!EXAMPLE 4(A) ‏ FINDING A NUMBER GIVEN ITS CIRCULAR FUNCTION VALUE

USE THE INVERSE COSINE FUNCTION OF A CALCULATOR.

APPROXIMATE THE VALUE OF S IN THE INTERVAL IF COS S = .9685.

, SO IN THE GIVEN INTERVAL, S ≈ .2517.


!EXAMPLE 4(B) ‏ FINDING A NUMBER GIVEN ITS CIRCULAR FUNCTION VALUE

RECALL THAT , AND IN QUADRANT III, TAN S IS NEGATIVE.

FIND THE EXACT VALUE OF S IN THE INTERVAL IF TAN S = 1.


CLASSWORK

FINISH WORKSHEETS

MEASURING ANGLES

RIGHT TRIANGLE TRIG

EXACT TRIG VALUES

3.1-3.2 WORKSHEET (BOTH SIDES)

FAMILIES OF TRIG FUNCTIONS


RADIAN “SNOWMEN”

LABEL THE SEVENTEEN POINTS(DON’T FORGET 360!) OF THE UNIT CIRCLE

DEGREES THEN RADIANS ON THE INSIDE

ON A SEPARATE PIECE OF PAPER, SHOW THE CONVERSIONS OF ALL SEVENTEEN POINTS!

WRITE EACH POINT AT EACH ANGLE ALONG THE UNIT CIRCLE ON THE OUTSIDE

COLOR AND DECORATE YOUR “SNOWMAN”


EXAMPLES


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