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§1.6 Trigonometric Review

The student will learn about:angles in degree and radian measure,trigonometric functions, graphs of sine and cosine functions, and the four other trigonometric functions.

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§1.6 Trigonometric Review

What follows is basic information about the trigonometric essentials you will need for calculus. It is not complete and assumes you have a full knowledge of trigonometric functions.

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Degrees and RadiansAngles are measured in degrees where there are 360º in a circle.Angles are also measured in radians where there are 2π radians in a circle.

Example 1. Convert 90º to radians.

Degree-Radian Conversion Formula

rad180raddeg

rad180

90rad

180

90rador

radrad

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Degrees and Radians

Example 2. Convert π/3 radians to degrees.

= 60 º

Degree-Radian Conversion Formula

rad180raddeg

rad

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180rad

rad

rad3

180or

Some Important AnglesRadian 0 π/6 π/4 π/3 π/2 π 2πDegree 0º 30º 45º 60º 90º 180º 360º

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Trigonometric FunctionsConsider a unit circle with center at the origin. Let point P be on the circle and form an angle of θ (in radians) with the positive x axis.

θ(1, 0)

(0, 1)P (x, y)

The sine θ is the ordinate of point p, i.e. y = sine θ.

The cosine θ is the abscissa of point p, i.e. x = cosine θ.

To find the value of either the sine or cosine functions use the sin and cos keys of your calculator. Make sure you are in the correct mode, [either degrees or radians] pertaining to the problem.

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Trigonometric FunctionsConsider a unit circle with center at the origin. Let point P be on the circle and form an angle of θ (in radians) with the positive x axis.

θ(1, 0)

(0, 1)P (x, y)

The sine θ is the ordinate of point p, i.e. y = sine θ.

The cosine θ is the abscissa of point p, i.e. x = cosine θ.

Remember the sign “+/-” of the abscissa and the ordinate in the different quadrants. That will help you get their signs correct in the future.

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Trigonometric FunctionsThe cosine θ is the abscissa of point p, i.e. x = cosine θ.The sine θ is the ordinate of point p, i.e. y = sine θ.

The sine and cosine of some special angles

Radian 0 π/6 π/4 π/3 π/2 π 2π

Degree 0º 30º 45º 60º 90º 180º 360º

Sine 0 1/2 2/2 3/2 1 0 0

Cosine 1 3/2 2/2 1/2 0 - 1 1

The values in aqua will repeat. Know them!

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Graphs of Sine and Cosinef (x) = sine x is a periodic function that repeats every 2π radians and can be found on your graphing calculator as:

f (x) = cosine x is a periodic function that repeats every 2π radians and can be found on your graphing calculator as:

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Graphs of Sine and Cosine

Being able to picture these graphs in my mind has helped me a lot in determining the numeric value of a trig function. If you combine this information with the basic numerical information given earlier you will be in pretty good shape for getting the correct numerical values for the trig functions.

y = sin x y = cos x

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Four Other Trig FunctionsFour Other Trigonometric Functions

These functions may also be graphed on your calculator.

0xcosxcos

xsinxtan

0xsinxsin

xcosxcot

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Four Other Trig FunctionsFour Other Trigonometric Functions

These functions may also be graphed on your calculator.

0xcosxcos

1xsec

0xsinxsin

1xcsc

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Trigonometric Identities

There are literally hundreds of trig identities. Several of the most useful follow.

Reciprocal identities

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Trigonometric Identities

Pythagorean Identities

sin 2 x + cos 2 x = 1

1 + tan 2 x = sec 2 x

1 + cot 2 x = csc 2 x

Quotient Identities

xcos

xsinxtan

cos xcot x

sin x

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Trigonometric Identities

Sum Angle Identities

ysinxcosycosxsin)yx(sin

cos(x y) cos xcos y sin xsin y

tan x tan ytan(x y)

1 tan xtan y

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Trigonometric Identities

Double Angle Identities

sin(2x) 2sin xcos x

2

2 tan xtan(2x)

1 tan x

2 2cos(2x) cos x sin x 22cos x 1

21 2sin x

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Trigonometric Identities

And lots more

Handout

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Summary.

We defined the six trigonometric functions.

We examined the graphs of the trigonometric functions.

We are now ready to continue our study of calculus using the trigonometric functions.

Degree-Radian Conversion Formula

rad180raddeg

We examined some trig identities.

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ASSIGNMENT

§1.6; Page 28; 1 – 11.