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Precalculus Core Concept Cheat Sheet

11: Radian, Degree Measure and the Unit Circle of Trigonometric Fun^ęy^

_Key Terms

•    Angle: An angle conasts of two rays, R_; and R;, and a common vertex. orien we think of an angle as being a rotation of R_; onto Rj. R_; iscalled the initial side and Rj is called terminal side. When the rotation is counterclockwise, the angle is positive. and when the rotation is clockwlse, the angle is negattve.

•    Standard positlon: Angle is sald to Ce In standard position, if its vertex at the ongm of a rectangular coordmate system and its inittal side comsides with the positive x-axis.

•    Cotcrminal anglcs: Two angles m standard posltions are said to t>e cotermrnal if they have the same initial side and terminal stde.

•    Dcgrcc: The angle formed by rotaing the initial side exactly once in the counterclockwise direction until It comcides with Itself (1 revolution) is said to measure 360¹. One degree, 1*, 1/360 revolution.

•    Central angle: is an angle whose vertex is at the center of a circle.

•    Quadrantal angle: the terminal side of the angle lies on the x-axts or y-axis.

•    Radian: If the radius of the circle is rand the length of arc subtendsed Dy the central angle is aiso r, then the measure of the angle is 1 radian.

•    Pcriodlc function: A function is called periodic <f there positive number p such that, whenever ne in the cfc<nsr./ is 0*p. and

f(0 +p)= f(0)._jjfl

Key Formulas

•    The arc length s:    5 = vO

Where r is the radius and O is angle Notę: O is measured in radian.

Basic Transformat»o.'t.

• Area of the sector:

I

A = ~r-ę.

Where r is the radius and O is .WJsi::? ’. Notę: O is also in radian.

•    Linear speed u of the rixiiagjnm

•    Angular speed

Słx trigo;3arne^»: ftirtcUc-iij

•    Sine funr.:-in •.:){/-

•    Cosinc    • v

%

• 1 counterclockwise revolubon = 360‘:*ć\ • v.:i ;iw . 1° = 60'

.    I'= 60“

•    1 degree = radian

IKO

ot.v:tfu{ e to evaluate the exact ^[lr.v‘ t->r;oriOmetric functions

ri^.v:i'..Ws^x,y) of the point on the unit >V>r.'i-: łi> tnat angle; i^fdrformular to get the values of the

tyj procedurę to using calculator to valucs of the trigonometric

functions._

< Siłę 1- S|):er the angle in the calculator using radian or

*Sjc-a‘/r Set the calculator to the correct MODĘ;

«..Si„5*. 3: Your calculator has the keys marked sin, cos and press the key to find the desired value;

■ Step 4: To find the vaiues of the remaining three trigonometric funebons: secant, cosecant and cotangent through the relations:

,, ^{1 1}

x cos 0

c*0-I—L

v sin 0

ci >10-i-. ¹

y tan 0

Problem Solving tips

Thoroughly read the entire problem.

Sketch the angle in a coordinate system ir needed.

Seek the coordinate of point P(x,y) on the terminal side. Use the defmltton of the trigonometric functions.

When solvmg the problem, keep the basie properties of the tngonometne fucbons In mind.

Do any mathematlcal calculations carefully.

fit V:GS^v: rhese are the keys related this topie. Try to read through it carefully twice then recite It out on a u-.-n-:. ~>nwv* it agam before the exams.