Thursday, May 2, 2013

Trigonometric Identities

Reciprocal identities

Quotient Identities

Co-Function Identities

Pythagorean Identities

Sum and Difference Formulas

Double Angle Formulas

Function	Abbreviation	Description	Trigonometric Identities using Radians
Sine	sin	$\frac {\textrm{opposite}} {\textrm{hypotenuse}}$	$\sin \theta = \cos \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\csc \theta}$
Cosine	cos	$\frac {\textrm{adjacent}} {\textrm{hypotenuse}}$	$\cos \theta = \sin \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\sec \theta}\,$
Tangent	tan	$\frac {\textrm{opposite}} {\textrm{adjacent}}$	$\tan \theta = \frac{\sin \theta}{\cos \theta} = \cot \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\cot \theta}$
Cotangent	cot	$\frac {\textrm{adjacent}} {\textrm{opposite}}$	$\cot \theta = \frac{\cos \theta}{\sin \theta} = \tan \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\tan \theta}$
Secant	sec	$\frac {\textrm{hypotenuse}} {\textrm{adjacent}}$	$\sec \theta = \csc \left(\frac{\pi}{2} - \theta \right) =\frac{1}{\cos \theta}$
Cosecant	csc	$\frac {\textrm{hypotenuse}} {\textrm{opposite}}$	$\csc \theta = \sec \left(\frac{\pi}{2} - \theta \right) =\frac{1}{\sin \theta}$

Non-permissible values

Sin x/Cos x Cos x cannot be equal to zero

Therefore the non-permissible values of Sin x/Cos x is where Cos x is equal to zero on the unit circle.

Cos x is equal to zero at [ (0,1) 90°   π/2 ] And [ (0,-1)   270° 3π/2   ]

general formula for non-permissible values: X ≠   π/2 +   πn , n  ∈ I

Tan x/Sec x Tan x/Csc x=   Sin x/Cos x / 1/Sin x

Cos x cannot be equal to zero

Sin x cannot be equal to zero

Therefore the non-permissible values of Tan x/Sec x is where cos x and sin x are equal to zero.

Cos x is equal to zero at [ (0,1) 90° π/2 ] And [ (0,-1) 270° 3π/2 ]

Sin x is equal to zero at [ (1,0) 0°, 360° 0,2π ] And [ (-1,0) 180° π ]

Cos x X ≠ π/2 + πn , n ∈ I

Sin x X ≠ πn , n ∈ I

general formula for non-permissible values: X ≠ π/2n , n ∈ I

Exact Trigonometric values for angles

2Sinπ/6cosπ/6 2sin α cos α = sin 2α

α = π/6 2sin α cos α = sin 2π/6

sin 2π/6= sin π/3

sin π/3 Sin 60° Sin= Opp/Hyp

Exact value: = √3 / 2

http://www.wolframalpha.com

Tuesday, April 23, 2013

Hey guys! shelly here to help YOU with transformation Sine and Cosine graphs (:

What you need to know:

One period is the length of one cycle in degrees or radians, the formula for period is 2pi / |b|

The amplitude is the highest or lowest point from the middle axis also known as |a|, the formula for amplitude is max-min / 2

Basic Sine Function

Basic Cosine Function

Now you know everything you need to know to graph a Sine or Cosine function, let's get started!

Example 1: graph the function y = cos ( x + 45) -2

This means that the function must move 45 degrees ( or pi/2 ) to the left and 2 units down

period: 2pi / |b| = 2pi / 1 = 2pi

amplitude = 1

phase shift: move pi / 2 to the left

Vertical Displacement: 2 units down

Example 2: y= sin (x + pi/2) + 1

The function must go pi/2 ( or 90 degrees) to the left and 1 unit up

period : 2pi / |b| = 2pi / 1= 2pi

amplitude: 1

phase shift : pi/2 to the left

vertical displacement : 1 units up

And WALLAH that's how you graph transformed sine and cosine functions! Hope this was helpful and simple enough for all of you to understand ! (:

Monday, April 8, 2013

Special Right Triangles

Hello guys :) This is chitwan and today's short blog is going to be about Special Right Triangles.

A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45–45–90. This is called an "angle-based" right triangle.

"Angle-based" special right triangles are specified by the relationships of the angles of which the triangle is composed. The angles of these triangles are such that the larger (right) angle, which is 90 degrees or π/2 radians, is equal to the sum of the other two angles.

The side lengths are generally deduced from the basis of the unit or other geometric methods. This approach may be used to rapidly reproduce the values of trigonometric functions for the angles 30°, 45°, and 60°.

Special triangles are used to aid in calculating common trigonometric functions, as below:

Degrees	Radians	sin	cos	tan
0	0	$\tfrac{\sqrt{0}}{2}=0$	$\tfrac{\sqrt{4}}{2}=1$	$0$
30	$\tfrac{\pi}{6}$	$\tfrac{\sqrt{1}}{2}=\tfrac{1}{2}$	$\tfrac{\sqrt{3}}{2}$	$\tfrac{1}{\sqrt{3}}$
45	$\tfrac{\pi}{4}$	$\tfrac{\sqrt{2}}{2}=\tfrac{1}{\sqrt{2}}$	$\tfrac{\sqrt{2}}{2}=\tfrac{1}{\sqrt{2}}$	$1$
60	$\tfrac{\pi}{3}$	$\tfrac{\sqrt{3}}{2}$	$\tfrac{\sqrt{1}}{2}=\tfrac{1}{2}$	$\sqrt{3}$
90	$\tfrac{\pi}{2}$	$\tfrac{\sqrt{4}}{2}=1$	$\tfrac{\sqrt{0}}{2}=0$	$\infty$

Let's look at the first of the special angle triangles, and this will hopefully become clear. Take a right angle triangle with two 45 degree angles, and with sides of 1 unit length. By the Theorem of Pythagoras, the hypotenuse of this triangle is of length √2. This is what this triangle looks like:

So then, from these values and SOHCAHTOA, you can obtain the trig values for this special angle of 45 degrees. You can see that:

Sin(45) = 1/√2
Cos(45) = 1/√2
Tan(45) = 1

Don't worry if you can't remember these exact ratios... the simplest thing to remember is how to construct the special angle triangle... which is as easy as remembering a right angle triangle with a 45 degree angle and 2 sides of length 1...

The second of the special angle triangles, which represents the rest of the special angles to remember, is slightly more complex, but still straightforward. Take a right angle triangle with angles of 30, 60, and 90 degrees. The simplest lengths of the sides of this triangle are 1, 2, √3 (with 2 being the longest side, the hypotenuse). This 30 60 90 triangle looks like this:

So then, using the these values, you can obtain the trig values for these special angles as well:

Sin(30) = 1/2
Cos(30) = √3/2
Tan(30) = 1/√3

Sin(60) = √3/2
Cos(60) = 1/2
Tan(60) = √3/1 = √3

And if you still don't get this or are dumb like me just go and look at the notes we did. Believe me they are much more simple and easier to understand :)

Here are some links that can help you with special right triangles: :p

http://www.youtube.com/watch?v=MHvHyv0aubw

http://www.youtube.com/watch?v=6Cb-XzSMXo4

Sunday, April 7, 2013

The Unit Circle

Hello guys, it's Renz and today's blog would be all about the Unit Circle.

Unit Circle

The "Unit Circle" is a circle with a radius of 1.

Being so simple, it is a great way to learn and talk about lengths and angles.

The center is put on a graph where the x axis and y axis cross, so we get this neat arrangement he

re.

Sine, Cosine and Tangent

Because the radius is 1, you can directly measure sine, cosine and tangent.

What happens when the angle, θ is 0°?

cos=1, sin=0 and tan=0

What happens when θ is 90°?

cos=0, sin=1 and tan is undefined

Pythagoras

Pythagoras' Theorem says that for a right angled triangle, the square of the long side equals the sum of the squares of the other two sides:

x² + y² = 1²

But 1² is just 1, so:

x² + y² = 1
(the equation of the unit circle)

Also, since x=cos and y=sin, we get:

(cos(θ))² + (sin(θ))² = 1
(a useful "identity")

Important Angles: 30°, 45° and 60°

You should try to remember sin, cos and tan for the angles 30°, 45° and 60°.

Yes, yes, it is a pain to have to remember things, but it will make life easier when you know them, not just in exams, but other times when you need to do quick estimates, etc.

These are the values you should remember!

Angle	Sin	Cos	Tan=Sin/Cos
30°			¹/_√3 = ^√3/₃
45°			1
60°			√3

ow To Remember?

To help you remember, think "1,2,3" :

sin(30°) = ^√1/₂ = ¹/₂ (because √1 = 1)
sin(45°) = ^√2/₂
sin(60°) = ^√3/₂

And cos goes "3,2,1"

cos(30°) = ^√3/₂
cos(45°) = ^√2/₂
cos(60°) = ^√1/₂ = ¹/₂ (because √1 = 1)

What about tan?

tan = sin/cos, so you can calculate:

tan(30°) =	sin(30°)	=	1/2	=	1

	cos(30°)		√3/2		√3

tan(45°) =	sin(45°)	=	√2/2	= 1

	cos(45°)		√2/2

tan(60°) =	sin(60°)	=	√3/2	= √3

	cos(60°)		1/2

Where did those values come from?

We can use the equation x² + y² = 1 to find the lengths of x and y (which are equal to cos and sin when the radius is 1):

45 Degrees

For 45 degrees, x and y are equal, so y=x:

x² + x² = 1

2x² = 1

x² = ½

x = y = √½

60 Degrees

Take an equilateral triangle (all sides are equal and all angles are 60°) and split it down the middle.

The "x" side is now ½,

And the "y" side will be:

(½)² + y² = 1

¼ + y² = 1

y² = 1-¼ = ¾

y = √¾

30 Degrees

30° is just 60° with x and y swapped, so x = √¾ and y = ½

Summary

√½ is usually changed to this:
And √¾ is usually changed to this:

And here is the result (as before):

Angle	Sin	Cos	Tan=Sin/Cos
30°			¹/_√3 = ^√3/₃
45°			1
60°			√3

Putting it All Together

And here they are for every quadrant. With the correct sign (plus or minus) as per Cartesian Coordinates.

Note that cos is first and sin is second, so it goes (cos, sin):

And this is the same Unit Circle in radians.

Here's a website that may help you as well:

https://www.khanacademy.org/math/trigonometry/basic-trigonometry/unit_circle_tut/e/unit_circle

Source:

http://www.mathsisfun.com/geometry/unit-circle.html