B Y: I S I S G I L L - R E I D

E S S E N T I A L S TA N D A R D S

LAW OF SINES: AMBIGUOUS LAW OF SINES

LAW OF SINES

T H E L A W O F S I N E S S TA T E S T H A T T H E R A T I O O F T H E L E N G T H O F A S I D E O F A T R I A N G L E T O T H E S I N E O F
THE ANGLE OPPOSITE THAT SIDE IS THE SAME FOR ALL SIDES AND ANGLES IN A GIVEN TRIANGLE. THE
AMBIGUOUS CASE IS IF TWO SIDES AND AN ANGLE OPPOSITE ONE OF THEM IS GIVEN, THREE POSSIBILITIES

DEFINITION

ONE SOLUTION EXISTS GIVEN A= 22, B=12 AND A=40°.
THE OTHER ANGLES AND SIDE.

EXAMPLE
By the Law of Sines,

B is acute.
C ≈ 180° – 40° –
20.52° ≈ 119.48°
By the Law of Sines,

FIND

+ W R I T E A N E Q U A T I O N I N V O LV I N G T H E U N K N O W N H E I G H T O F T H E T R E E .
1.

FIND THE HEIGHT OF THE TREE.

R E A L L I F E S I T U AT I O N

P Y T H A G O R E A N I D E N T I T Y, R E C I P R O C A L I D E N T I T Y, D O U B L E A N G L E F O R M U L A S
FOR SINE AND COSINE, HALF ANGLE FORMULAS FOR SINE AND COSINE

PROVING TRIGONOMETRIC
IDENTITIES

A N I D E N T I T Y I S A N E Q U A T I O N W H I C H I S A LW A Y S T R U E . F O R E X A M P L E ,
PYTHAGOREAN THEOREM'S (BASE) 2 + (PERPENDICULAR) 2 = (HYPOTENUSE) 2

DEFINITION

EXAMPLE
PYTHAGOREAN THEOREM'S (BASE) 2 + (PERPENDICULAR) 2 = (HYPOTENUSE) 2
IS TRUE FOR RIGHT TRIANGLES.
THERE ARE SIX BASIC TRIGONOMETRIC FUNCTIONS AND THEIR RECIPROCALS ARE
G I V E N B E L O W.
1. S I N X = 1 C S C X
2. C O S X = 1 S E C X
3. TA N X = 1 C O T X
4. C O T X = 1 TA N X
GIVEN BELOW ARE THE PYTHAGOREAN IDENTITIES.
1. S I N 2 X + C O S 2 X = 1
2. 1 + TA N 2
3. 1 + C O T 2

X = SEC 2

X = CSC 2

X

X

FOR THE BASIC TRIGONOMETRIC FUNCTIONS, GIVEN BELOW ARE THE DOUBLE ANGLE

F O R M U L A S F O R S I N , C O S A N D TA N .
SIN 2X = 2 SINX COSX
COS 2X =
COS 2 X−SIN 2 X2COS 2 X−11−2SIN 2 X

TA N 2 X = 2 TA N X 1 − TA N 2 X
GIVEN BELOW ARE THE HALF ANGLE FORMULA'S FOR BASIC TRIGONOMETRIC FUNCTIONS.
SIN (X2 ) = ± 1−COSX2 − − − − − √
COS(X2 )= ± 1+COSX2 − − − − − √
TA N ( X 2 ) =
±1−COSX1+COSX − − − − − √ SINX1+COSX
1−COSXSINX

LIGHT WAVES ARE GRAPHS OF