Important Angles

Learn to Love Radians0° = 0 Radians45° =90° =135° =180° =225° =270° =315° =360° = 2π Radians

π Radians

π/2 Radians

3π/2 Radians

π/4 Radians

3π/4 Radians

5π/4 Radians

7π/4 Radians

30° =

60° =

π/6 Radians

π/3 Radians

sin (nπ/2) where n = any integerTo find the sine of intervals of π/2 (90°) you need to either use the unit circle where sin x is the y-value or graph y = sin x

0 1 .5 7 0 7 9 6 3 2 6 7 9 4 9 3 .1 4 1 5 9 2 6 5 3 5 8 9 8 4 .7 1 2 3 8 8 9 8 0 3 8 4 7 6 .2 8 3 1 8 5 3 0 7 1 7 9 6

-1

0

1

-1 0 1

-1

0

1

(0,1) π/2

(1,0) 0, 2π

(–1,0) π

(0,–1) 3π/20 π/2 π 3π/2 2π

0 1 .5 7 0 7 9 6 3 2 6 7 9 4 9 3 .1 4 1 5 9 2 6 5 3 5 8 9 8 4 .7 1 2 3 8 8 9 8 0 3 8 4 7 6 .2 8 3 1 8 5 3 0 7 1 7 9 6

-1

0

1

cos (nπ/2) where n = any integerTo find the sine of intervals of π/2 (90°) you need to either use the unit circle where sin x is the x-value or graph y = cos x

-1 0 1

-1

0

1

(0,1) π/2

(1,0) 0, 2π

(–1,0) π

(0,–1) 3π/20 π/2 π 3π/2 2π

sin and cos of nπ/2sin (0) = cos (0) =sin (π/2) = cos (π/2) =sin (π) = cos (π) =sin (3π/2) = cos (3π/2) = 0 1 .5 7 0 7 9 6 3 2 6 7 9 4 9 3 .1 4 1 5 9 2 6 5 3 5 8 9 8 4 .7 1 2 3 8 8 9 8 0 3 8 4 7 6 .2 8 3 1 8 5 3 0 7 1 7 9 6

-1

0

1

-1 0 1

-1

0

1

(0,1) π/2

(1,0) 0, 2π

(–1,0) π

(0,–1) 3π/2

0 π/2 π 3π/2 2π

0 1 .5 7 0 7 9 6 3 2 6 7 9 4 9 3 .1 4 1 5 9 2 6 5 3 5 8 9 8 4 .7 1 2 3 8 8 9 8 0 3 8 4 7 6 .2 8 3 1 8 5 3 0 7 1 7 9 6

-1

0

1

0 π/2 π 3π/2 2π

010–1

10–10

6 Trig FunctionsHow do you calculate the each trig functionsin θ =cos θ =tan θ =sec θ =csc θ =cot θ =

opposite / hypotenuseadjacent / hypotenuseopposite / adjacenthypotenuse / adjacenthypotenuse / oppositeadjacent / opposite

Don’t memorize sin π/3, cos π/6, tan π/4, etc.Memorize the 2 triangles and then use your knowledge of trig to figure out sin, cos, tan, etc.

π/2 π/3

π/6

x√3 2xπ/2 π/4

π/4

x

x x√2

π/4-π/4-π/2 Triangle

π/2 π/4

π/4

x

x x√2

sin (π/4) =

cos (π/4) =

tan (π/4) =

sec (π/4) =

csc (π/4) =

cot (π/4) =

π/6-π/3-π/2 Triangle

π/2 π/3

π/6

x

x√3 2x

sin (π/6) =

cos (π/6) =

tan (π/6) =

sec (π/6) =

csc (π/6) =

cot (π/6) =

π/6-π/3-π/2 Triangle

π/2 π/3

π/6

x

x√3 2x

sin (π/3) =

cos (π/3) =

tan (π/3) =

sec (π/3) =

csc (π/3) =

cot (π/3) =

Inverse Trig FunctionsWorking backwards, find the angleRemember, sin-1 x means sin of ___ angle = xsin-1 0 = sec-1 1 =sin-1 0.5 = sec-1 √2 =sin-1 1 = csc-1 √2 =cos-1 1 = tan-1 0 =cos-1 0.5 = tan-1 1 =cos-1 0 = tan-1 √3 =