T r i g o n o m e t r y T r i v i u m

[Pages:5]Trigonometry Trivium

Main Trig Identities

cos2 x + sin2 x = 1

cos2 x - sin2 x = cos 2x, 2 sin x cos x = sin 2x

cos2 x

=

1

+ cos 2x ,

sin2 x = 1 - cos 2x

2

2

cos(-x) = cos x, sin(-x) = - sin x, tan(-x) = - tan x, cot(-x) = - cot x

cos(x + y) = cos x cos y - sin x sin y, sin(x + y) = sin x cos y + cos x sin y

sin x

1 cos x

tan x = , cot x =

=

cos x

tan x sin x

tan x + tan y tan(x + y) =

1 - tan x tan y

1

1

1

sec x =

, csc x =

=

cos x

sin x sec x

Examples

1.

Solve

the

equation:

sin x

=

1 2

Solution: From the unit circle we get two solutions inside the interval [0, 2):

5

x1 = 6

and x2 =

. 6

Sine is a periodic function with the period 2. So, the actual solutions are made of two infinite sequences of numbers

5 5

5

5

5

, + 2, - 2, + 4, - 4, ... and , + 2, - 2, + 4, - 4, ...

66

6

6

6

66

6

6

6

5

It

briefly can

be

written

as

x1

=

+ 2k, 6

x2

=

+2k, where k is an integer number (positive 6

or negative).

5

Notice that

6

=

-

. 6

Then

x2

=

- 6

+

and

these

two

sequences

of

solutions

can

be

1

combined into a single formula x = (-1)k + k. 6

Answer: x = (-1)k + k. 6

2. Solve the equation: cos x =

3 2

Solution:

From

the

unit

circle

we

get

two

solutions

x1

=

6

and

x2

=- . 6

Cosine

is

a

periodic

function with the period 2. So, the actual solutions are made of two infinite sequences of

numbers

, + 2, - 2, + 4, - 4, ... and - , - + 2, - - 2, - + 4, - - 4, ...

66

6

6

6

66

6

6

6

It

briefly

can

be

written

as

x1

=

6

+ 2k,

x2

=

- 6

+ 2k.

These two sequences of solutions can be combined in a single formula x = ? + 2k.

6

Answer: x = ? + 2k.

6

3.

Solve

the

equation:

sin x

=

-

1 3

Solution: From the unit circle we get two solutions between 0 and 2:

x1 = sin-1

-

1 3

= - sin-1

1 3

and x2 = sin-1

1 3

+ .

The full set of solutions is given by x1 = - sin-1

1 3

+ 2k and x2 = sin-1

1 3

+ + 2k.

These two sequences of solutions can be combined into a single formula

x = (-1)k-1 sin-1

1 3

+ k.

Answer: x = (-1)k-1 sin-1

1 3

+ k.

4.

Solve

the

equation:

cot x - sin x

=

2 sin2

x 2

Solution:

2 sin2

x 2

=

1 - cos x.

Then

cos x

cot x - sin x = 1 - cos x

+ cos x - sin x - 1 = 0

sin x

cos x + cos x sin x - sin x(sin x - 1)

=0

sin x

cos x(1 + sin x) - sin x(sin x + 1) = 0 and sin x = 0

2

 (cos x - sin x)(sin x + 1) = 0 and x = n cos x = sin x or sin x = -1 and x = n

x = + k or x = - + 2k and x = n.

4

2

Answer:

x1 = 4 + k,

x2

=

- 2

+ 2k.

5. Prove the identity: cos 2x + 1 = 2 cos2 x Solution: cos 2x = cos2 x - sin2 x, 1 = cos2 x + sin2 x. Then cos 2x + 1 = cos2 x - sin2 x + cos2 x + sin2 x = 2 cos2 x.

6. Prove the identity: tan x + cot x = sec x csc x

Solution:

sin x cos x sin2 x + cos2 x

1

11

tan x + cot x =

+

=

=

=

?

= sec x csc x.

cos x sin x cos x sin x cos x sin x cos x sin x

7. Find the domain of the function: f (x) =

3

(a) when - x

2

2

5

(b) when x

2

2

(c) when 0 x 2

1 2

-

sin

x

Solution:

The

domain

can

be

found

from

the

inequality

1 2

- sin x

0

or

sin x

1 2

.

7

From the unit circle we get the set of intervals of the solution: x - + 2k, + 2k

6

6

7 (a) Take k = 0 to get the answer x - ,

66 5 13 (b) Take k = 1 to get the answer x , 66

(c) Here we take a combination of solutions from parts (a) and (b) to make a solution that lies inside the interval [0, 2].

5

The answer is x 0, , 2

6

6

3

Exersices. Calculators are NOT permitted.

Easy

Prove the identities

1. cos x tan x = sin x

tan x

2.

= sin x

sec x

3. (1 - tan x)2 = sec2 x - 2 tan x

4. (cos x - sin x)2 = 1 - sin 2x

cos x

5.

= tan x + sec x

1 - sin x

Solve the equations. Find all solutions.

2

6. sin x = 2

7. tan x = 3

3

8. sin x = - 2

10. sin 2x cos 2x = 0 11. sin 2x = cos 2x.

1 9. cos 3x = -

2

Medium

Prove the identities

12.

1

1

+

= 2 csc2 x

1 - cos x 1 + cos x

13. (cos x - sin x)2 + (cos x + sin x)2 = 2

14. cot2 x - cos2 x = cot2 x cos2 x

1 + cos x sec x + 1

15.

=

1 - cos x sec x - 1

Solve the equations. Find all solutions.

2

16. sin x cos x = 4

17. 4 cos2 x - 4 cos x + 1 = 0

19. sin4 4x - cos4 4x = 1

1 + tan2 x 20. sin2 x + cos2 x = 2

18. 3 sin x = 2 cos2 x

cot 2x

2

21.

cos2 x - sin2 x

=

. 3

Hard (challenge) problems

Prove the identities

22.

sin 9x + sin 10x + sin 11x + sin 12x

=

4 cos

x 2

cos x sin

21 2

x

23.

cos 2x - cos 3x - cos 4x + cos 5x

=

-4 sin

x 2

sin x cos

7 2

x

4

24. sin2 2x - cos

3

-

2x

sin

2x

-

6

1 =

4

tan 3x 1 - cot2 3x 25. tan2 3x - 1 ? cot 3x = 1

26. 3 - 4 cos 2x + cos 4x = tan4 x 3 + 4 cos 2x + cos 4x

Solve the equations

27. cos 3x - sin x = 3 (cos x - sin 3x)

k

Ans: x1 = 8 + 2 , x2 = 12 + k

28.

1

4 cot x + cot2

x

+

sin2

2x

+

1

=

0

Ans: x = - + k

4

29.

sin 2x

=

cos4

x 2

- sin4

x 2

Ans:

x1

=

2

+ k,

x2

=

(-1)k

6

+ k

30. (1 + cos 4x) sin 2x = cos2 2x

Ans:

x1

=

4

+

k ,

2

x2

=

(-1)k

12

+

k 2

31. tan x - tan 2x = sin x Ans: x = k

32.

3 sin2 x + 3 cos2 x =

34

k Ans: x = +

42

In all answers k is an integer number.

Find the domain of the functions

33. f (x) =

cos

x

-

1 2

when

- x

2

2

34. f (x) =

3 2

-

cos

x

when

0 x 2

3

35. f (x) = 2 - tan x when x 0, ,

2

22

5

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