exact trigonometry tables

exact trigonometry tables

0.1

Basic angles

Since the trigonometric ratios for most angles cannot be
calculated exactly in closed algebraic form,
a few well-known angles
that can be calculated
often comprise the bulk of textbook
exercises involving

trigonometry

.

The basic angles are given in Table

1

.

Table 1:

Basic angles encountered in trigonometry

θ

sin

⁡

θ

cos

⁡

θ

tan

⁡

θ

0

∘

0

1

0

30

∘

1/2

3

/

2

1

/

3

45

∘

2

/

2

2

/

2

1

60

∘

3

/

2

1/2

3

90

∘

1

0

∞

0.2

Other angles by addition and halving

These basic angles can be easily extended to obtain more angles of interest. Adding

multiples

of

90

∘

merely rotates these angles into other

quadrants

; the appropriate values of

sin

and

cos

can be obtained through

symmetry

.

The values for

15

∘

can be obtained by using the formula for the

difference

of angles (

http://planetmath.org/AngleSumIdentity

):

sin

⁡

15

∘

=

sin

⁡

(

45

∘

-

30

∘

)

=

sin

⁡

45

∘

⁢

cos

⁡

30

∘

-

cos

⁡

45

∘

⁢

sin

⁡

30

∘

=

2

2

⋅

3

2

-

2

2

⋅

1

2

=

6

-

2

4

.

Likewise, we can find that

cos

⁡

(

15

∘

)

=

6

+

2

4

sin

⁡

(

75

∘

)

=

sin

⁡

(

45

∘

+

30

∘

)

=

6

+

2

4

cos

⁡

(

75

∘

)

=

6

-

2

4

.

More exact angles can be obtained by solving the

double angle identity

:

sin

⁡

θ

2

=

±

1

-

cos

⁡

θ

2

,

cos

⁡

θ

2

=

±

1

+

cos

⁡

θ

2

.

So for example,

sin

⁡

7.5

∘

=

(

4

-

6

-

2

)

/

8

. These angles can be further added and subdivided to obtain a dense subset of exactly known angles. However, such effort is not generally useful. Computers and calculators use a

combination

of lookup-tables and numeric iteration to obtain their values.

0.3

The angles

18

∘

,

36

∘

,

54

∘

,

72

∘

The

18

∘

-

36

∘

-

54

∘

-

72

∘

series of angles cannot be obtained
by halving, doubling, adding or subtracting the previous angles. Nevertheless, they are

constructible

, and their exact values can be derived by the following elementary procedure:

Consider an

isosceles triangle

with the angles

72

∘

,

54

∘

and

54

∘

.
From the

triangle

we derive the

relation

:

sin

⁡

72

∘

2

=

cos

⁡

54

∘

Notice that

72

=

4

×

18

and

54

=

3

×

18

, so if

x

=

18

∘

,
then

sin

⁡

2

⁢

x

=

cos

⁡

3

⁢

x

2

⁢

sin

⁡

x

⁢

cos

⁡

x

=

4

⁢

cos

3

⁡

x

-

3

⁢

cos

⁡

x

2

⁢

sin

⁡

x

=

4

⁢

cos

2

⁡

x

-

3

2

⁢

sin

⁡

x

=

4

⁢

(

1

-

sin

2

⁡

x

)

-

3

The last

equation

is a quadratic equation that can be solved for

sin

⁡

18

∘

. Carrying out the calculations, we obtain the values in Table

2

.

Table 2:

Other

constructible angles

in trigonometry

θ

sin

⁡

θ

cos

⁡

θ

18

∘

5

-

1

4

5

+

5

2

⁢

2

36

∘

5

-

5

2

⁢

2

5

+

1

4

54

∘

5

+

1

4

5

-

5

2

⁢

2

72

∘

5

+

5

2

⁢

2

5

-

1

4

Title

exact trigonometry tables

Canonical name

ExactTrigonometryTables

Date of creation

2013-03-22 15:31:26

Last modified on

2013-03-22 15:31:26

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

11

Author

stevecheng (10074)

Entry type

Example

Classification

msc 51M04

Classification

msc 33B10

Related topic

ConstructibleAnglesWithIntegerValuesInDegrees

Related topic

Cotangent2

Related topic

Trigonometry

Related topic

TheoremOnConstructibleAngles

Related topic

GraphOfEquationXyConstant

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