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Trigonometry

Trigonometry begins in the right triangle, but it doesn’t have to be restricted to triangles. The trigonometric functions carry the ideas of triangle trigonometry into a broader world of real-valued functions and wave forms.

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Trigonometry Topics

Radian Measure The Unit Circle Trigonometric Functions Larger Angles Graphs of the Trig Functions Trigonometric Identities Solving 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.

rs sr

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Radian Measure

degrees

360

radians

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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 Problems Find the degree

measure equivalent of radians.

degrees

360

radians

210

360

r

2

2360 420

420

360

7

6

r

r

degrees

360

radians

360

3 4

2

22 270

135

d

d

d

3

4

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 Circle

Connect 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

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

x

y1

is the angle of rotation

xx

1

cosyy

1

sin

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The Unit Circle

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.

yy

1

)sin(

xx

1

)cos(

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Trigonometric Functions

sin( ) y

x

y1

is the angle of rotation

x

y)tan(

y

1)csc(

x)cos(x

1)sec(

y

x)cot(

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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 II

Original 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 III

Original 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 angle

For 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 Seniors Take Calculus Use the phrase “All Seniors Take Calculus”

to remember the signs of the trig functions in different quadrants.

AllSeniors

Take Calculus

All functions are positive

Sine is positive

Tan is positive Cos is positive

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Angles measured in degrees:

1sin 45 cos45 and tan 45 1

2

Angles measured in radians:

1sin / 4 cos / 4 and tan / 4 1

2

Special Right Triangles

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Angles measured in degrees:

1sin30 cos60

2

3sin 60 cos30

21

tan 60 3tan30

Special Right Triangles

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The 16-Point Unit Circle

0

1/2

2 /2

3 /2

1

1

3 /2

2 /2

1/2

0

0

3 /3

1

3

2

2

2 3 /3

1

1

2 3 /3

2

2

3

1

3 /3

0

3 /2

2 /2

1/2

0

1/2

2 /2

3 /2

1

3

1

3 /3

0

2 3 /3

2

2

2

2

2 3 /3

1

3 /3

1

3

1/2

2 /2

3 /2

1

3 /2

2 /2

1/2

0

3 /2

2 /2

1/2

0

1/2

2 /2

3 /2

1

3 /3

1

3

3

1

3 /3

0

2

2

2 3 /3

1

2 3 /3

2

2

2 3 /3

2

2

2

2

2 3 /3

1

3

1

3 /3

0

3 /3

1

3

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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

khxbay )(sin

2b

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Sample Problem Which 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)

-2p -1p 1p 2p

5

4

3

2

1

-1

-2

-3

-4

-5

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Sample Problem Find the baseline between

the high and low points. Graph is translated -1

vertically. Find height of each peak.

Amplitude is 3 Count number of waves in

2 Frequency is 2

-2p -1p 1p 2p

5

4

3

2

1

-1

-2

-3

-4

-5

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

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Graphs of the Trig Functions

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

Amplitude 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

khxbay )(cos

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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

-2p -1p 1p 2p

8

6

4

2

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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

-2p -1p 1p 2p

8

6

4

2

y = 5 cos(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 Functions Secant 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

xx

sec

1cos

tansin

cosx

x

x

cotcos

sinx

x

x

Quotient Identities

xx

cot

1tan

xx

csc

1sin

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Trigonometric Identities

Cofunction Identities The function of an angle = the

cofunction of its complement.

)90cot(tan xx

)90csc(sec xx

)90cos(sin xx

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Trigonometric Identities

Pythagorean Identities

The fundamental

Pythagorean identity

Divide the first by sin2x

Divide the first by cos2x xx 22 sec1tan

xx 22 csccot1

1cossin 22 xx

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Trigonometric Identities

2 2

2

2

2

cos2 cos sin

cos2 1 2sin

cos2 2cos 1

sin 2 2sin cos

2 tantan 2

1 tan

2sin cos sin( ) sin( )

2cos cos cos( ) cos( )

2sin sin cos( ) cos( )

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Trigonometric Identitiescos( ) cos cos sin sin

cos( ) cos cos sin sin

sin( ) sin cos cos sin

sin( ) sin cos cos sin

tan tantan( )

1 tan tan

tan tantan( )

1 tan tan

-

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Solving Trig Equations Solve 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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Sample Problem

Solve

90or150,30 xx

3 3 2 02 sin cosx x

1sinor2

1sin xx

0cos2sin33 2 xx

0)sin1)(sin21( xx

0sin2sin31 2 xx

0)sin1(2sin33 2 xx

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All these relationships are based on the assumption that the triangle is a right triangle.

It is possible, however, to use trigonometry to solve for unknown sides or angles in non-right triangles.

Law of Sines and Cosines

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In geometry, you learned that the largest angle of a triangle was opposite the longest side, and the smallest angle opposite the shortest side.

The Law of Sines says that the ratio of a side to the sine of the opposite angle is constant throughout the triangle.

a

A

b

B

c

Csin( ) sin( ) sin( )

Law of Sines

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In ABC, mA = 38, mB = 42, and BC = 12 cm. Find the length of side AC. Draw a diagram to see the position of the given

angles and side. BC is opposite A You must find AC, the side opposite B.

A B

C

Law of Sines

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.... Find the length of side AC. Use the Law of Sines with mA = 38, mB = 42, and

BC = 12

a

A

b

Bsin( ) sin( )

12

38 42sin( ) sin( ) b

38sin42sin12 b

38sin

42sin12b

041.13042.13

029.8

Law of Sines

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WarningWarning

The Law of Sines is useful when you know the sizes of two sides and one angle or two angles and one side.

However, the results can be ambiguous if the given information is two sides and an angle other than the included angle (ssa).

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Law of Cosines

If you apply the Law of Cosines to a right triangle, that extra term becomes zero, leaving just the Pythagorean Theorem.

The Law of Cosines is most useful when you know the lengths of all three sides and need to

find an angle, or when you two sides and the included angle.

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Triangle XYZ has sides of lengths 15, 22, and 35. Find the measure of the angle C.

c a b ab C

C

C

C

2 2 2

2 2 2

2

35 15 22 2 15 22

1225 225 484 660

1225 709 660

cos( )

cos( )

cos( )

cos( )

15 22

35

C

Law of Cosines

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... Find the measure of the largest angle of the triangle.

516 660

516

6607818

7818 14141

cos( )

cos( ) .

cos ( . ) .

C

C

C

15 22

35

Law of Cosines

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Laws of Sines and Cosines

a

b

c

B

C

A

Cabbac

C

c

B

b

A

a

cos2

sinsinsin

222

Law of Sines:

Law of Cosines: