Trigonometry

Preparing for the SAT II

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Trigonometry

Trigonometry begins in the right Trigonometry begins in the right triangle, but it doesn’t have to be triangle, but it doesn’t have to be

restricted to triangles. The restricted to triangles. The trigonometric functions carry the trigonometric functions carry the

ideas of triangle trigonometry into a ideas of triangle trigonometry into a broader world of real-valued broader world of real-valued functions and wave forms. functions and wave forms.

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Trigonometry Topics

Radian MeasureThe Unit CircleTrigonometric FunctionsLarger AnglesGraphs of the Trig FunctionsTrigonometric IdentitiesSolving Trig Equations

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Radian Measure

To talk about trigonometric functions, it is helpful to move to a different system of angle measure, called radian measure.

A radian is the measure of a central angle whose intercepted arc is equal in length to the radius of the circle.

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Radian Measure

degrees360

radians

2

There are 2 radians in a full rotation -- once around the circle

There are 360° in a full rotation To convert from degrees to radians or

radians to degrees, use the proportion

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Sample ProblemsFind the degree

measure equivalent of radians.

degrees360

radians

210360

r

2

2360 420

420360

76

r

r

degrees360

radians

3603 4

2

22 270

135

d

d

d

34

Find the radian measure equivalent of 210°

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The Unit Circle

Imagine a circle on the coordinate plane, with its center at the origin, and a radius of 1.

Choose a point on the circle somewhere in quadrant I.

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The Unit CircleConnect the origin to the

point, and from that point drop a perpendicular to the x-axis.

This creates a right triangle with hypotenuse of 1.

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The Unit Circle

sin( ) y y1

cos bg x x1

x

y1

is the angle of rotation

The length of its legs are the x- and y-coordinates of the chosen point.

Applying the definitions of the trigonometric ratios to this triangle gives

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The Unit Circle

sin( ) y y1

cos bg x x1

The coordinates of the chosen point are the cosine and sine of the angle . This provides a way to define functions sin()

and cos() for all real numbers .

The other trigonometric functions can be defined from these.

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Trigonometric Functions

sin( ) y

cos bgx

tan bg yx

csc bg1y

sec bg1x

cot bg xy

x

y1

is the angle of rotation

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Around the Circle

As that point moves around the unit circle into quadrants II, III, and IV, the new definitions of the trigonometric functions still hold.

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Reference Angles

The angles whose terminal sides fall in quadrants II, III, and IV will have values of sine, cosine and other trig functions which are identical (except for sign) to the values of angles in quadrant I.

The acute angle which produces the same values is called the reference angle.

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Reference Angles

The reference angle is the angle between the terminal side and the nearest arm of the x-axis.

The reference angle is the angle, with vertex at the origin, in the right triangle created by dropping a perpendicular from the point on the unit circle to the x-axis.

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Quadrant IIOriginal angle

Reference angle

For an angle, , in quadrant II, the reference angle is

In quadrant II, sin() is positive cos() is negative tan() is negative

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Quadrant IIIOriginal angle

Reference angle

For an angle, , in quadrant III, the reference angle is -

In quadrant III, sin() is negative cos() is negative tan() is positive

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Quadrant IV

Original angle

Reference angleFor an angle, , in

quadrant IV, the reference angle is 2

In quadrant IV, sin() is negative cos() is positive tan() is negative

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All Star Trig Class

Use the phrase “All Star Trig Class” to remember the signs of the trig functions in different quadrants.

AllStar

Trig Class

All functions are positive

Sine is positive

Tan is positive Cos is positive

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Sine The most fundamental sine wave, y=sin(x),

has the graph shown. It fluctuates from 0 to a high of 1, down to –1,

and back to 0, in a space of 2.

Graphs of the Trig Functions

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The graph of is determined by four numbers, a, b, h, and k. The amplitude, a, tells the height of each peak and

the depth of each trough. The frequency, b, tells the number of full wave

patterns that are completed in a space of 2. The period of the function is The two remaining numbers, h and k, tell the

translation of the wave from the origin.

Graphs of the Trig Functions

y a b x h k sin b gc h

2b

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Sample ProblemWhich of the following

equations best describes the graph shown? (A) y = 3sin(2x) - 1 (B) y = 2sin(4x) (C) y = 2sin(2x) - 1 (D) y = 4sin(2x) - 1 (E) y = 3sin(4x)

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Sample ProblemFind the baseline between the

high and low points. Graph is translated -1

vertically.Find height of each peak.

Amplitude is 3Count number of waves in 2

Frequency is 2

y = 3sin(2x) - 1

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Cosine The graph of y=cos(x) resembles the graph of

y=sin(x) but is shifted, or translated, units to the left.

It fluctuates from 1 to 0, down to –1, back to 0 and up to 1, in a space of 2.

Graphs of the Trig Functions

2

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Graphs of the Trig Functions

y a b x h k cos b gc hAmplitude a Height of each peakFrequency b Number of full wave patterns Period 2/b Space required to complete waveTranslation h, k Horizontal and vertical shift

The values of a, b, h, and k change the shape and location of the wave as for the sine.

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Which of the following equations best describes the graph? (A) y = 3cos(5x) + 4 (B) y = 3cos(4x) + 5 (C) y = 4cos(3x) + 5 (D) y = 5cos(3x) +4 (E) y = 5sin(4x) +3

Sample Problem

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Find the baseline Vertical translation + 4

Find the height of peak Amplitude = 5

Number of waves in 2 Frequency =3

Sample Problem

y = 5cos(3x) + 4

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Tangent The tangent function has a

discontinuous graph, repeating in a period of .

Cotangent Like the tangent, cotangent is

discontinuous. • Discontinuities of the cotangent

are units left of those for tangent.

Graphs of the Trig Functions

2

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Graphs of the Trig FunctionsSecant and Cosecant

The secant and cosecant functions are the reciprocals of the cosine and sine functions respectively.

Imagine each graph is balancing on the peaks and troughs of its reciprocal function.

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Trigonometric Identities

An identity is an equation which is true for all values of the variable.

There are many trig identities that are useful in changing the appearance of an expression.

The most important ones should be committed to memory.

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Trigonometric Identities

Reciprocal Identities

sincsc

xx

1

cossec

xx

1

tancot

xx

1

tan sincos

x xx

cot cossin

x xx

Quotient Identities

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Cofunction Identities The function of an angle = the cofunction of its

complement.

Trigonometric Identities

sin cos( )x x 90

sec csc( )x x 90

tan cot( )x x 90

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Trigonometric Identities

sin cos2 2 1x x

1 2 2 cot cscx xtan sec2 21x x

Pythagorean Identities The fundamental Pythagorean identity

Divide the first by sin2x Divide the first by cos2x

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Solving Trig EquationsSolve trigonometric equations by following

these steps: If there is more than one trig function, use

identities to simplify Let a variable represent the remaining function Solve the equation for this new variable Reinsert the trig function Determine the argument which will produce the

desired value

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Solving Trig Equations

To solving trig equations:

Use identities to simplify

Let variable = trig function

Solve for new variable

Reinsert the trig function

Determine the argument

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Solve Use the Pythagorean

identity • (cos2x = 1 - sin2x)

Distribute Combine like terms Order terms

Sample Problem

3 3 2 0

3 3 2 1 0

3 3 2 2 0

1 3 2 0

2 3 1 0

2

2

2

2

2

sin cos

sin sin

sin sin

sin sin

sin sin

x x

x x

x x

x x

x x

c h

3 3 2 02 sin cosx x

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Let t = sin x

Factor and solve.

Sample ProblemSolve 3 3 2 02 sin cosx x

2 3 1 02sin sinx x 2 3 1 02 1 1 0

2 1 0 1 02 1 1

12

2t tt t

t tt t

t

( )( )

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Sample ProblemSolve 3 3 2 02 sin cosx x

x 6

56

or

x 2

x 6

56 2

, ,

Replace t = sin x. t = sin(x) = ½ when t = sin(x) = 1 when

So the solutions are