Docsity
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Docsity AI
Docsity AI
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

Trigonometry summary sheet, Summaries of Trigonometry

Academy of Careers and TechnologyTrigonometry

Defining relations for tangent, cotangent, secant, and cosecant in terms of sine and cosine. The Pythagorean formula for sines and cosines Periodicity of trig functions Identities for negative angles The Pythagorean formula for tangents and secants Identities expressing trig functions in terms of their supplements. Sum, difference, and double angle formulas for tangent. Triple angle formulas More half-angle formulas Ptolemy’s identities, the sum and difference formulas for sine and cosine. ...

Typology: Summaries

2019/2020

Available from 02/22/2022

mitterand-deguenon
mitterand-deguenon 🇺🇸

1 document

1 / 1

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
QUADRANTS & SIGNS OF F UNCTIONS
___________________________________________________________________________
RIGHT-ANGLE TRIANGLE RELATIONSHIPS
a = opposite side
b = adjacent side
c = hypotenuse
Pythagorean theorem:
___________________________________________________________________________
FUNDAMENTAL IDENTITIES
1. 2. 3. 4.
5. 6. 7.
8a. 8b.
9a. 9b.
10.
OPPOSITE-ANGLE IDENTITIES
1. 2. 3.
4. 5. 6.
ADDITION LAWS
1. 2.
3.
DOUBLE-ANGLE IDENTITIES
1a. 1b. 1c.
1d. 1e.
2. 3.
HALF-ANGLE IDENTITIES
1.
2.
3a.
3b.
PRODUCT IDENTITIES
1. 2.
3. 4.
5.
SUM IDENTITIES
1. 2.
3. 4.
___________________________________________________________________________
REDUCTION IDENTITY
1. , where is chosen so that
and
___________________________________________________________________________
PLANE TRIANGLE RELATIONSHIPS
Law of sines:
Law of cosines: ,
Law of tangents:
___________________________________________________________________________
INVERSE TRIGONOMETRIC FUNCTIONS
PRINCIPAL VALUES FOR INVERSE TRIGONOMETRIC FUNCTIONS
INVERSE IDENTITIES (ASSUMING PRINCIPAL VALUES ARE USED)
1. 7.
2. 8.
3. 9.
4. 10.
5. 11.
6. 12.
___________________________________________________________________________
COMPLEX IDENTITIES
Euler Identity:
___________________________________________________________________________
QUADRATIC FORMULA
Solution to :
___________________________________________________________________________
EXACT VALUES
III
III IV
All p ositive
sin & csc pos.
others neg.
cos & sec pos.
others neg.
tan & co t pos.
others neg.
θ
θθ
θ
90° - θ
θθ
θ
a
b
c
90°
θsin side opposite θ
hypotenuse
------------------------------------ a
c
---
==
θcos side adjacent to θ
hypotenuse
------------------------------------------b
c
---
==
θtan side opposite θ
side adjacent to θ
------------------------------------------a
b
---
==
c2a2b2
+=
xtan xsin
xcos
-----------
=xsec 1
xcos
-----------
=xcsc 1
xsin
----------
=xcot 1
xtan
-----------xcos
xsin
-----------
==
x
2
cos x
2
sin+1=1x
2
tan+x
2
sec=x
2
cot 1+x
2
csc=
π2⁄x–()cos xsin=π2⁄x+()cos xsin–=
π2⁄x–()sin xcos=π2⁄x+()sin xcos=
π2⁄x–()tan xcot=
x–()cos xcos=x–()sin xsin–= x–()tan xtan–=
x–()sec xsec=x–()csc xcsc–= x–()cot xcot–=
xy±()cos xycoscos xysinsin
+
−
=xy±()sin xycossin xysincos±=
xy±()tan xtan ytan±
1xytantan
+
−
-------------------------------
=
2xcos x
2
cos x
2
sin–=2xcos 2 x
2
cos 1–=2xcos 1 2 x
2
sin–=
x
2
sin 1
2
---12xcos–()=x
2
cos 1
2
---12xcos+()=
2xsin 2 xxcossin=2xtan 2xtan
1x
2
tan–
---------------------
=
x
2
---cos 1xcos+2
---------------------±=+ if x/2 is in quadrant I or IV
- if x/2 is in quadrant II or I II
x
2
---sin 1xcos–2
--------------------±=+ if x/2 is in quadrant I or II
- if x/2 is in quad rant III or IV
x
2
---tan 1xcos–
1xcos+
---------------------±=+ if x/2 is in quadrant I or III
- if x/2 is in quadrant II or I V
x
2
---tan 1xcos–xsin
-------------------- xsin
1xcos+
---------------------xcsc xcot–===
2xcos ycos xy–()cos xy+()cos+=2xsin ysin xy–()cos xy+()cos–=
2xsin ycos xy+()sin xy–()sin+=2xysincos xy+()sin xy–()sin–=
mxcos nxcos mn+()xcos mn–()xcos+=
xcos ycos+2
xy+
2
-----------


cos xy–
2
-----------


cos=xcos ycos–2
xy+
2
-----------


sin xy–
2
-----------


sin–=
xsin ysin+2
xy+
2
-----------


sin xy–
2
-----------


cos=xsin ysin–2
xy–
2
-----------


sin xy+
2
-----------


cos=
axsin bxcos+a2b2
+xy+()sin=y
ycos a
a2b2
+
--------------------
=ysin b
a2b2
+
--------------------
=
Function Domain Range Quadrants
I and II
I and IV
all reals I and IV
or , I and II
or , I and IV
all reals I and II
Principal values for Principal values for
Angle (deg) Angle (rad) cos(x) sin(x) tan(x)
100
1
01
-1 0 0
0-1
A
B
C
a
b
c
a
Asin
----------- b
Bsin
----------- c
Csin
------------
==
c2a2b22ab Ccos–+=
Ca2b2c2
–+
2ab
---------------------------


acos=
ab+
ab–
------------
1
2
---AB+()tan
1
2
---AB–()tan
------------------------------
=
yx
1–
cos=1– x1≤≤ 0yπ≤≤
yx
1–
sin=1– x1≤≤ π2⁄–yπ2⁄≤≤
yx
1–
tan=π2⁄–yπ2⁄<<
yx
1–
sec=x1≥x1–≤0yπ≤≤ yπ2⁄≠
yx
1–
csc=x1≥x1–≤π2⁄–yπ2⁄≤≤ y0≠
yx
1–
cot=0yπ<<
x0≥x0<
0x
1–
sin π2⁄≤≤ π2⁄–x
1–
sin 0<≤
0x
1–
cos π2⁄≤≤ π2⁄x
1–
cos π≤<
0x
1–
tan π2⁄<≤π2⁄–x
1–
tan 0<<
0x
1–
cot π2⁄≤<π2⁄x
1–
cot π<<
0x
1–
sec π2⁄<≤π2⁄x
1–
sec π≤<
0x
1–
csc π2⁄≤<π2⁄–x
1–
csc 0<≤
x
1–
sin x
1–
cos+π2⁄=x–()
1–
sin x
1–
sin–=
x
1–
tan x
1–
cot+π2⁄=x–()
1–
cos πx
1–
cos–=
x
1–
sec x
1–
csc+π2⁄=x–()
1–
tan x
1–
tan–=
x
1–
csc 1 x⁄()
1–
sin=x–()
1–
cot πx
1–
cot–=
x
1–
sec 1 x⁄()
1–
cos=x–()
1–
sec πx
1–
sec–=
x
1–
cot 1 x⁄()
1–
tan=x–()
1–
csc x
1–
csc–=
wcos eiw eiw–
+
2
---------------------
=iwsin eiw eiw–
–
2
--------------------
=eiw wcos iwsin+=
ax2bx c++ 0=xb–b24ac–±2a
--------------------------------------
=
0°0
30°π6⁄32⁄12⁄33⁄
45°π4⁄22⁄22⁄
60°π3⁄12⁄32⁄3
90°π2⁄∞±
180°π
270°3π2⁄∞±
TRIGONOMETRY REVIEW SHEET
___________________________________________________________________________

Partial preview of the text

Download Trigonometry summary sheet and more Summaries Trigonometry in PDF only on Docsity!

Q UADRANTS & S IGNS OF F UNCTIONS

___________________________________________________________________________

R IGHT-A NGLE TRIANGLE RELATIONSHIPS

a = opposite side b = adjacent side c = hypotenuse

Pythagorean theorem:

F UNDAMENTAL IDENTITIES

8a. 8b.

9a. 9b.

O PPOSITE -A NGLE I DENTITIES

A DDITION L AWS

D OUBLE -ANGLE I DENTITIES

1a. 1b. 1c.

1d. 1e.

H ALF-A NGLE I DENTITIES

3a.

3b.

P RODUCT I DENTITIES

S UM I DENTITIES

___________________________________________________________________________

R EDUCTION I DENTITY

  1. , where is chosen so that

and

___________________________________________________________________________

P LANE TRIANGLE R ELATIONSHIPS

Law of sines:

Law of cosines: ,

Law of tangents:

___________________________________________________________________________

I NVERSE TRIGONOMETRIC FUNCTIONS

P RINCIPAL VALUES FOR I NVERSE TRIGONOMETRIC FUNCTIONS

I NVERSE I DENTITIES ( ASSUMING PRINCIPAL VALUES ARE USED )

___________________________________________________________________________

C OMPLEX I DENTITIES

Euler Identity:

Q UADRATIC FORMULA

Solution to :

EXACT VALUES

II I

III IV

All positive

sin & csc pos.

others neg.

cos & sec pos.

others neg.

tan & cot pos.

others neg.

θθθθ

90° - θθθθ a

b

c

90°

sin θ side opposite^ θ hypotenuse

------------------------------------ a c

cos θ= side adjacent to----------------------------------------- hypotenuse^ θ-^ = bc^ ---

tan θ side opposite^ θ side adjacent to θ

------------------------------------------ a b

c^2 = a^2 + b^2

tan x sin x cos x

= ----------- sec x^1 cos x

= ----------- csc x^1 sin x

= ---------- cot x^1 tan x

----------- cos x sin x

cos^2 x + sin^2 x = 1 1 + tan^2 x = sec^2 x cot 2 x + 1 =csc^2 x cos ( π ⁄ 2 – x )= sin x cos ( π ⁄ 2 + x )=–sin x sin ( π ⁄ 2 – x )= cos x sin ( π ⁄ 2 + x )=cos x tan( π ⁄ 2 – x )=cot x

cos( – x )= cos x sin( – x )= – sin x tan ( – x )=–tan x sec ( – x )= sec x csc ( – x )= – csc 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

tan ( x ± y ) tan^ x ±tan y 1 +−tan x tan y

cos 2 x = cos^2 x – sin^2 x cos 2 x = 2 cos 2 x – 1 cos 2 x = 1 – 2 sin^2 x

sin 2 x^1 2

= --- 1( – cos 2 x ) cos 2 x^1 2

=--- 1( +cos 2 x )

sin 2 x = 2 sin x cos x tan 2 x^2 tan x 1 – tan^2 x

x 2

cos ---^1 +cos x 2

  • if x/2 is in quadrant I or IV - if x/2 is in quadrant II or III

x 2

sin ---^1 – cos x 2

  • if x/2 is in quadrant I or II - if x/2 is in quadrant III or IV

x 2

tan ---^1 – cos x 1 +cos x

  • if x/2 is in quadrant I or III - if x/2 is in quadrant II or IV

x 2

tan-- - = 1 -------------------^ – sincos xx -^ = 1 -------------------- +sincos xx -^ =csc x – cot x

2 cos x cos y = cos( x – y )+cos( x + y ) 2 sin x sin y =cos ( x – y )–cos( x + y ) 2 sin x cos y = sin ( x + y )+sin( x – y ) 2 cos x sin y =sin ( x + y )–sin( x – y ) cos mx cos nx =cos ( m + n ) x +cos( m – n ) x

cos x + cos y 2 x^ + y 2

cos  ----------- x^ – y 2

= cos  ----------- cos x – cos y 2 x^ + y 2

sin  ----------- x^ – y 2

=– sin  -----------

sin x + sin y 2 x^ + y 2

sin (^)  ----------- x^ – y 2

= cos  ----------- sin x – sin y 2 x^ – y 2

sin  ----------- x^ + y 2

= cos  -----------

a sin x + b cos x = a^2 + b^2 sin( x + y ) y

cos y a a^2 + b^2

= -------------------- sin y b a^2 + b^2

Function Domain Range Quadrants I and II

I and IV

all reals I and IV

or , I and II

or , I and IV

all reals I and II

Principal values for Principal values for

Angle (deg) Angle (rad) cos(x) sin(x) tan(x) 1 0 0

1

0 1

-1 0 0

0 -

A

B

C

a

b

c

a sin A

----------- b sin B

----------- c sin C

c^2 = a^2 + b^2 – 2 ab cos C

C a

(^2) + b (^2) – c 2 2 ab =acos ^ -------------------------- - 

a + b a – b

tan--- ( A + B )

1 2 tan---^ ( A – B )

y = cos–^1 x –^1 ≤^ x^ ≤^10 ≤^ y^ ≤π

y = sin–^1 x –^1 ≤^ x^ ≤^1 –^ π^ ⁄^2 ≤^ y^ ≤π^ ⁄^2

y = tan–^1 x –^ π^ ⁄^2 <^ y^ <π^ ⁄^2

y = sec–^1 x x^ ≥^1 x^ ≤^ –^10 ≤^ y^ ≤^ π y^ ≠π^ ⁄^2

y = csc–^1 x x^ ≥^1 x^ ≤^ –^1 –^ π^ ⁄^2 ≤^ y^ ≤^ π^ ⁄^2 y^ ≠^0

y = cot–^1 x^0 <^ y^ <π

x ≥ 0 x < 0

0 ≤ sin –^1 x ≤ π ⁄ 2 – π ⁄ 2 ≤sin –^1 x < 0

0 ≤ cos –^1 x ≤ π ⁄ 2 π ⁄ 2 <cos –^1 x ≤π

0 ≤ tan– 1 x <π ⁄ 2 – π ⁄ 2 < tan –^1 x < 0

0 < cot– 1 x ≤π ⁄ 2 π ⁄ 2 < cot –^1 x <π

0 ≤ sec –^1 x <π ⁄ 2 π ⁄ 2 <sec –^1 x ≤π

0 < csc –^1 x ≤π ⁄ 2 – π ⁄ 2 ≤csc –^1 x < 0

sin –^1 x + cos–^1 x = π ⁄ 2 sin –^1 (– x )=–sin–^1 x tan –^1 x + cot–^1 x = π ⁄ 2 cos– 1 (– x )=π – cos–^1 x sec –^1 x + csc–^1 x = π ⁄ 2 tan –^1 ( – x )=–tan–^1 x csc –^1 x = sin–^1 ( 1 ⁄ x ) cot –^1 (– x )=π – cot–^1 x sec –^1 x = cos–^1 ( 1 ⁄ x ) sec –^1 (– x )=π – sec–^1 x cot –^1 x = tan–^1 ( 1 ⁄ x ) csc– 1 (– x )=–csc–^1 x

cos w e^

iw (^) + e – iw 2

= --------------------- i sin w e^

iw (^) – e – iw 2

= -------------------- e iw^ =cos w + i sin w

ax^2 + bx + c = 0 x –^ b b ±^2 – 4 ac 2 a

=^ --------------------------------------

30 ° π ⁄ (^63) ⁄ 2 1 ⁄ (^23) ⁄ 3

45 ° π ⁄ (^42) ⁄ 2 2 ⁄ 2

60 ° π ⁄ 3 1 ⁄ (^23) ⁄ 2 3

90 ° π ⁄ 2 ±∞

180 ° π

270 ° 3 π ⁄ 2 ±∞

TRIGONOMETRY R EVIEW SHEET

___________________________________________________________________________