Trigonometry Week 1, Test 1

Cos(pi/6)

sqrt(3)/2 /

Tan 3pi/2

undefined ( -1/0)

Since (x, y) are on the circle of radius r, we know that

x^2 + y^2 = r^2

Criteria for similarity

(Angle, Angle)= If two corresponding angles are congruent, then the triangles are similar, (Side, Angle, Side)= Two angles have equal ratio, both triangles share an equal angle (Side, Side, Side) = If all sides share similar ratios (A/a = B/b = C/c) then triangles are similar, angles are equal

(x,y) on the terminal side

(Cos(x), Sin(x))

Tan ( pi/3)

(sqrt (3)/2) / one half Tan (pi3) = sqrt (3)

On a Triangle drawn on a Cartesian plane unit circle The side adjacent to acute angle alpha is = to coordinate point

(x) = Cos (alpha)

On a Triangle drawn on a Cartesian plane unit circle The side opposite to acute angle alpha is = to coordinate point

(y) = Sin (alpha)

Cos (pi), Cos (180)

-1

Sin 3pi/2

-1

Cos( 2pi/3)

-1/2

cos( 5pi/6)

-sqrt(3)/2

Cos 3(pi)/2

0

Sin (pi), Sin (180)

0

Sin 0 = Sin 2 pi

0

Tan (pi)

0

Cos(90 and pi/2)

0 (x coordinate)

Cos 0 = Cos 2pi

1

Tan (pi/4)

1

Sin( 90 and pi/2)

1 (y coordinate)

Sin^2 =

1-Cos^2(alpha)

Cos ^2 =

1-sin^2(alpha)

Cos (pi/3)

1/2

Sin(5pi/6)

1/2

Sec(x)

1/cos(x)

Csc(x)

1/sin(x)

2pi/3

120 degrees

5pi/6

150 degrees

Half Turn in Degrees

180 Degrees

Cos, Sin, Csc. Sec repeat every =

2Pi

360

2pi

Full Angle in Radians

2pi

Full Angle in Degrees

360 degrees

270

3pi/2

Quarter turn in Degrees

90 Degrees

Similar Triangle ratios

A/a = B/b = C/c

Cos(x)

Adjacent/Hypotenuse

Cot(x)

Adjacent/Opposite

Angles and Arcs formula

Alpha/360 = Length of Arc (s) divided by C (circumference of circle)

Linear Velocity

As the object moves around the circle, it does so with certain speed, this speed is the linear velocity, or Distance (s) divided by Time (t)

Formula for circumference

C= radius times angle alpha, when alpha is in radians

Cos (alpha) + 2 pi = Sin (alpha) +2 pi =

Cos (alpha) Sin (alpha) Why? Because of periodicity. The period of sine and cosine (and any trigonometric function) is 2pi (180 degrees)

Cot(x)

Cos x/ Sin x

Half Turn in Radians

Pi

What is the period of Tangent?

Pi

180 d

Pi radians

45

Pi/ 4

Quarter turn in Radians

Pi/2

D to R series 90 d

Pi/2 radians

60

Pi/3

30

Pi/6

On a Triangle drawn on a Cartesian plane unit circle The hypotenuse is =

Radius (r)

Tan^2 +1 =

Sec ^2 alpha

Tan(x)

Sin x/ Cos x y/x on Cartesian plane

Symmetries

Sin(-alpha) = -sin(alpha), odd Cos(-alpha) = cos (alpha), Even Tan (-alpha) = -tan (alpha) odd

Cos(pi/4)

Sqrt(2)/2

Sin (2pi/3)

sqrt(3)/2

sin (pi/6)

sqrt(3)/2

Converting from Degree Decimal to DMS

First number before decimal = Degree remaining decimal gets multiplied by 60, numbers before decimals are Minutes Remaining decimal gets multiplied by 60, remaining numbers are seconds

Tan(pi/2)

Undefined, as Tan = Y/X = Sin/Cosine, which of pi/2 = 1/0

Angle Pi/4

When on a plane, forms an isosceles triangle, with sides measuring H =1, x= sqrt(2)/2, y = sqrt(2)/2

On a Cartesian plane triangle with radius present (hypotenuse) x = y= hypotenuse=

X = r times cos alpha y = r times sin alpha Hypotenuse = r

Pythagorean Theorem (Cartesian plane)

a squared + b squared = r squared

Pythagorean Theorem

a²+b²=c²

(x,y) on the terminal side when radius r is present

Cos(x)/r, Sin(x)/r

Pythagorean theorem in terms of sin(x) cos (x)

Cos^2 (alpha) + Sin ^ (alpha) = 1

Fundamental identities

Csc = 1/sin Sec = 1/cos Tan = Sin/cos Cot = 1/tan = cos/sin

Converting from DMS to Degree Decimal

D + M/60 + S/3600

Area of a triangle in a Cartesian plane

Formula = A= 1/2 r^2 times angle alpha (in radians)

Converting Degrees to Radians

Formula: Alpha measured in degrees/ 360 = Alpha measured in Radians/ pi

Sec(x)

Hypotenuse/ Adjacent

Csc(x)

Hypotenuse/Opposite

Tan (0), Tan (2pi)

Opposite/ adjacent, which on a plan is equal to Y/X, which is Sin alpha/ Cos alpha. Sin0 = 0, Cos0= 1, so Tan0 = 0/1 = 0

Tan(x)

Opposite/Adjacent

Sin(x)

Opposite/Hypotenuse

Sin( pi/4)

Sqrt(2)/2

Sin ( pi/3)

Sqrt(3)/2

Angular Velocity

The rate at which the angle changes is called the angular velocity or (Radius (r) times Angle Alpha) divided by Time (t)

Tangent repeats every =

pi

tan (pi/6)

sqrt(3)/3

Supplementary

sum of angles that equals 180 degrees

Complementary

sum of angles that equals 90 degrees