Texas Instruments Trigonometry with the TI-84 Plus Silver Edition, Trigonometric Identities

Trigonometry with the TI-84 Plus Silver Edition

Trigonometric Functions

Special Values of Trigonometric Functions

Acute Angles

òsin ￿ = opp/hyp ôcos ￿ = adj/hyp ötan ￿ = opp/adj

Arbitrary Angles sin ￿ = b/r cos ￿ = a/r

tan ￿ = b/a

Real Numbers

sin t = y

cos t = x tan t = y/x

csc ￿ = hyp/opp sec ￿ = hyp/adj cot ￿ = adj/opp

csc ￿ = r/b sec ￿ = r/a cot ￿ = a/b

csc t = 1/y sec t = 1/x cot t = x/y

hyp

opp

u

adj

y

(a,b)

r	u
	x

y

(x,y)

t

(1,0)

x

t radians

￿	￿		sin ￿				cos ￿			tan ￿			cot ￿		sec ￿		csc ￿
(degrees) (radians)
0°	0		0				1			0			–		1		–
	π						√			√					2√
			1								3
									3							3
30°													√3				2
	6			2			2			3					3
		π	√				√2
						2
45°										1			1		√2		√2
		4	2				2
		π	√							√			√				2√
					3			1						3				3
60°												3			2
	3		2				2						3				3
90°		π	1				0			–			0		–		1
	2

Inverse Trigonometric Functions

Function	Domain		Range
y?y = sin-1x	–1 ≤ x ≤ 1	–		π	≤ y	≤	π
			2
	–1 ≤ x ≤ 1					2
y@y = cos-1x		0 ≤ y ≤ π

Special Triangles

√2

4π =45 ° 1

1	π =30°
	6

π=60°

3

2

1

√3

yAy = tan-1x

All real numbers

–

π

< y <

π

2

2

y = cot-1x

All real numbers

0 < y < π

≥ 1

π

∪

π

y = sec

-1

x

x

0,

,π

2

2

π

y = csc-1x

≥ 1

∪ 0,

π

x

–

, 0

2

2

Trigonometric Identities

csc t = 1/sin t			tan t = sin t/cos t
sec t = 1/cos t			cot t = cos t/sin t
cot t = 1/tan t
sin2 t + cos2 t = 1			sin (–t) = –sin t
1 + tan2 t = sec2 t			cos (–t) = cos t
1 + cot2 t = csc2 t			tan (–t) = –tan t
sin (u + v) = sin u cos v + cos u sin v
cos (u + v) = cos u cos v – sin u sin v
tan (u + v) =		tan u + tan v
	1 – tan u tan v

sin (u – v) = sin u cos v – cos u sin v cos (u – v) = cos u cos v + sin u sin v

tan (u – v) = tan u – tan v 1 + tan u tan v

sin 2u =2 sin u cos u

cos 2u = cos2 u – sin2 u = 1 – 2 sin 2 u = 2 cos2 u – 1

tan 2u =

2 tan u

1 – tan2 u

u

√

u

√

1 – cos u

sin

=

cos

1 + cos u

=

2

2

2

2

tan

u

=

1 – cos u=

sin u

tan2 u =

1 - cos 2u

2

1 + cos 2u

sin u

1 + cos u

sin2 u =

1 – cos 2u

cos2 u =

1 + cos 2u

2

2

sin u cos v

=

1/

2

[sin (u + v) + sin (u – v)]

1/

cos u sin v

=

2

[sin (u + v) – sin (u – v)]

1/

cos u cos v =

2

[cos (u + v) + cos (u – v)]

1/

sin u sin v

=

2

[cos (u – v) – cos (u + v)]

© Texas Instruments, 2007

Laws of Sines and Cosines

Sine								Cosine
sin A = sin B = sin C								a2 = b2 + c2 – 2bc cos A
a	b				c			c2 = a2 + b2 – 2ab cos C
								b2 = c2 + a2 – 2ac cos B
Unit Circle								90°
					120°				60°
			135°		(-		√3	(0,1)	√3		45°
						1		1
						2	, 2 )	( 2	, 2 )
	150°		(- √22 ,√22)						(	√22 ,√22)			30°
	( -	√3		1						(	√3		1
		2 ,		2 )							2	,	2 )
180°	(-1,0)											(1,0)		0°
	√3			1						(	√3		1
	( - 2		, -	2 )							2	, - 2 )
	210°		( -	√2	√2				(	√2	√2		330°
				2	, - 2 )					2 , -	2 )
	225°				(- 21 , -√23)			( 21 , -√23)			315°
					240°			(0,-1)	300°
								270°
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