A Review of Trigonometric

Functions

oppo

site

adjacent

hypotenuse

hypotenuse

a

b

c

opposite

adja

cent

Right Triangle Vocabulary

A

B

C

Trigonometric Functions Defined in terms of right triangles

sin(x) = opp/hypcos(x) = adj/hyptan(x) = opp/adj

= sin(x) / cos(x)

Know the graphs

Trigonometric Functions Defined in terms of the unit circle

1

P(x)=(cos x, sin x)

cos x

sin x

x

1

Other Trig Functions cot(x) = 1/tan(x) = cos(x) / sin(x) sec(x) = 1/cos(x) csc(x) = 1/sin(x)

Odd/Even Odd

Sin(x) Csc(x) Tan(x) Cot(x)

Even Cos(x) Sec(x)

1

C

Radians Radian measure of the

angle at the center of a unit circle equals the length of the arc that the angle cuts from the unit circle.

C

s

r

sr =

1

Radians

1

sr=

Note: Radian measureis a dimensionlessnumber

Radians and Degreess

2 r=2

radian measure2

arclengthcircumference

degree measure360°= =

2 = 360° = 180°

/2

/2

/2

/2

/2

Famous ValuesAngle0º = 0

30º = /6

45º = /4

60º = /3

90º = /2

Sin0

1

2

3

4

/2

/2

/2

/2

/2

Cos4

3

2

1

0

=0

=1/2

=1

=1

=1/2

=0

Tan0

1/ 3

1

3

Und

Domain, Range, Period

sin(x)

cos(x)

tan(x)

Domain(-, )

(-, )

x /2,3/2, ...

Range(-1, 1)

(-1, 1)

(-, )

Period2

2

Finding the Period Sin (3πx/2 + 4) Set term that includes x equal to the

period of the trig function3πx/2 = 2π

Solve for xx = 4/3 = Period

Trig Identities to Know Pythagorean Identities

sin2 x + cos2 x = 1 tan2 x + 1 = sec2 x cot2 x + 1 = csc2 x

Double Angle sin(2x)=2sin(x)cos(x) cos(2x)=cos2(x) – sin2(x)

Square sin2(x) = (1 – cos(2x))/2 cos2(x) = (1 + cos(2x))/2

Creating Inverse Trig Functions The trig functions are not 1-1 Restrict their domains

y = sin(x) -π/2 ≤ x ≤ π/2 y = cos(x) 0 ≤ x ≤ πy = tan(x) -π/2 < x < π/2

The Inverse Trig Functions y = sin-1 xor y = arcsin(x)

Domain: [-1, 1]Range: [-π/2, π/2]

y = cos-1 x or y = arccos(x)Domain: [-1, 1]Range: [0, π]

y = tan-1 x or y = arctan(x)Domain: (-∞, ∞)Range: (-π/2, π/2)

Examples sin x = 0.455

x = sin-1 (0.455) cos x = π/2

x = cos-1 (π/2) tan x = 8

x = tan-1 (8)