Trigonometry and Unit Circle

30-60-90

x, x√3, 2x

45-45-90

x, x, x√2

Sin

Opp/hyp

Cosine

adjacent/hypotenuse

Tan

Opp/adj

Adjacent Comp Angles

x+y=90

Sin(x)

cos(90-x)

cos(x)

Sin(90-x)

90

Π/2 radians

360

2Π radians

270

3Π/2 radians

180

Π radians

If x+y=90

sinx=cosx

(+,+)

Quad 1

(-,+)

Quad 2

(-,-)

Quad 3

(+,-)

Quad 4

coterminal angles

θ + 360n, share terminal side, same trig values

Reference triangle

Terminal side of the angle + perpendicular side dropped from terminal point

30-60-90 (r=1)

1/2, √3/2, 1

45-45-90 (r=1)

√2/2, √2/2, 1

Reference angle (Quad 2)

θr = 180 - θ

Reference angle (Quad 1)

θ=θr

Reference angle (Quad 3)

θr = θ - 180

Reference angle (Quad 4)

θr = 360 - θ

Positive angle

Couterclockwise from standard position

Negative angle

Clockwise from standard position

Angle of elevation

the angle between the HORIZONTAL and the LINE OF SIGHT when an observer is looking UP

Angle of Depession

the angle between the HORIZONTAL and the LINE OF SIGHT when an observer is looking DOWN

30

π/6

60

π/3

45

π/4

Sin π/6

1/2

Cos π/6

√3/2

Sin 5π/6

1/2

cos 5π/6

-√3/2

\sin\frac{11\pi}{6}

-1/2

Cos\frac{11\pi}{6}

\frac{\left(\sqrt3\right)}{2}

Sin 7π/6

-1/2

\cos\left(\frac{7\pi}{6}\right)

-√3/2

Sin π/4

√2/2

Cos π/4

√2/2

Sin 3π/4

√2/2

Cos 3π/4

-√2/2

Sin 5π/4

-√2/2

Cos 5π/4

-√2/2

Sin 7π/4

-√2/2

Cos 7π/4

√2/2

Sin π/3

√3/2

Cos π/3

1/2

Sin 2π/3

√3/2

Cos 2π/3

-1/2

Sin 4π/3

-√3/2

Cos 4π/3

-1/2

Sin 5π/3

-√3/2

Cos 5π/3

1/2

Csc x

1/y

Sec x

1/x

Cot x

x/y

sin θ

y/r

cos θ

x/r

tan θ

y/x

csc θ

r/y

sec θ

r/x

cot θ

x/y

r =

√x² + y²

All

cos + sec, sin + csc, tan + cot

Students (Sin)

-(cos + sec), -(tan + cot), sin + csc

Take (Tan)

-(cos + sec), tan + cot, -(sin + csc)

Calculus

cos + sec, -(tan + cot), -(sin + csc)