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Review of Trigonometry

Appendix D.3

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After this lesson, you should be able to:

work in radian measurefind reference anglesuse and recreate the unit circle to find trig values of special anglesrecognize and sketch the graphs of sine, cosine and tangentuse the Pythagorean trig identities and reciprocal identities to simplify trig expressionssolve basic trig equations

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Anglesinitial ray on x-axis

acute angles angles between 0 and /2 radians

obtuse angles angles between /2 and radians

co-terminal angles angles that share the same terminal ray

Ex: /2 and -3/2

initial ray

terminal ray

Standard position of an angle

(0, 0)x

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Measuring Angles

Positive angles measured counterclockwiseNegative angles measured clockwise

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Radian MeasureRadian measure of a central angle in the unit circle is the length of the arc of the sector.

r = 1

Unit circle

r

s = r

circle with radius r

The length of the sector

s r1s

Arc Length is

s

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Definitions of Trig Functions

x

y

r

(x,y) 22 yxr

Circular Function Definitions

r

ysin

r

xcos

x

ytan

y

rcsc

x

rsec

y

xcot

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Quadrant Signs for Trig Functions

Quad I: All trig functions are +

Quad II: Sine and cosecant are +

Quad III: Tangent and cotangent are +

Quad IV: Cosine and secant are +

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Common 1st Quadrant Angles

Degrees

0° 30° 45° 60° 90°

Radians

Sin Cos Tan

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Unit Circle Function Definitions

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Unit circle

x

yysin

xcos

x

ytan

y

1csc

x

1sec

y

xcot

r = 1

0°

360 °

30 °

45 °

60 °

330 °

315 °

300 °

120 °

135 °

150 °

240 °

225 °

210 °

180 °

90 °

270 °

For a positive angle.

r = 1

Unit Circle with Special Angles

Remember: x = cos, y = sin

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Reciprocal Identities

1sec

cos

1csc

sin

1cot

tan

1sin

csc

1cos

sec

1tan

cot

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Trigonometric Identities & Formulas

2 2sin cos 1 2 2cos 1 sin

2 2tan 1 sec

sin 2 2sin cos 2 2cos 2 cos sin

2 1cos (1 cos 2 )

2

2 1sin (1 cos 2 )

2

2 2sin 1 cos

2 2cot 1 csc

Note: Those written in blue should be memorized.

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Graph of SineGraph the function y = sin x over the interval [-2, 2]. State its amplitude, period,domain and range.

x

y

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Graph of CosineGraph the function y = cos x over the interval [-2, 2]. State its amplitude, period,domain and range.

x

y

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Graph of TangentGraph the function y = tan x over the interval [-2, 2]. State its period,domain and range.

x

y

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Practice with Conversions

Example: Convert -34/15 to degree measure.

Example: Convert 850° to exact radian measure.

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Practice with Trig FunctionsExample: Given a point on the terminal side of in standard position, find the exact value of the six trig. functions of .

P (-4, -3)

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Practice with Trig FunctionsExample: Given the quadrant and one trigonometric function value of in standard position, find the exact value of the other five trig. functions.A. Quadrant I; tan =

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Practice with Trig Functions

B. Quadrant III; cot = 1

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Solving Basic Trig EquationsExample 1Solve the equation without using a calculator. 3

cos 0 22

for

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Solving Basic Trig EquationsExample 2Solve the equation without using a calculator. 22sin 1 sin 0 2for

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Homework

Exercises for Appendix D.3: #1-7 all, 11-19 all, 27-35 odd

Appendix D.3 can be found online at the textbook site and also on the CD provided with your text.