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