Trevor Pythagoras Maths » pythagoras

Trigonometry Identities

admin — Tue, 23 Dec 2008 15:09:09 +0000

There a number of “identities” in trigonometry that can be found from the basic ideas of sin, cos and tan as explained in my earlier post. These identities can help in solving equations involving trig functions, especially when there are 2 or more different functions as the often allow you to write the equation in terms of one function, eg sin, that you can then solve. One of the identities is: sin² + cos² = 1. To prove this consider a right angled triangle with side a,b and c as shown below

From this we can use Pythagoras theorem to say: a²+b²=c^{2^{now we know sin t = b/c so b = csin t cos t = a/c so a = ccos t}} substituting these values in the above equation we get (csint)² +(ccost)² = c² canceling the c² we get sint² + cost² = 1

There are trig functions that are equal to 1 over sin, cos and tan called cosec = 1/sin, sec = 1/cos and cot = 1/tan. These can be remembered using the third letter rule as the third letter of each of these corresponds to the the function it is one over.

Using these a cos² + sin² = 1 we can calculate other identities tan²t + 1 = sec²t We can obtain this by dividing through by cos² as we know sin/cos = tan, cos/cos = 1 and 1/cos = sec. Other similar identities can be obtained for cosec and cot.

Also see

Proof of Cosine Rule (4)
Understand the Sine and Cosine Rules (11)
Sin, Cos and Tan (15)
Proof of Pythagorases Theroem (0)
Sine and Cos Graphs Differentiating sin and cos (7)
Area of a Triangle (3)

Proof of Pythagorases Theroem

admin — Mon, 20 Oct 2008 20:04:15 +0000

Pythagoras theorem, that the square of the longest side of a right angled triangle is equal to the sum of the squares of the other 2 sides.

Square used to prove Pythagoras theorem

This can be proved quite easily by drawing a square into which fit 4 of the same right angled triangle as shown below

As you can see the area of the whole square is equal to the the sum of the 2 shorter sides squared or (a+b)². The area of the green square left is the square of the longest side c². We also know that the area of each of the triangles is 1/2 x base x height = ab/2

From these 3 areas we can prove the theorem. The know that the total area of the square is equal to the area of the green square plus 4 of the triangles ie)

(a+b)²=c²+ 4ab/2
a² + b² + 2ab = c²+ 2ab

The 2ab ‘s cancel and we are left with Pythagoras theorem
a² + b² = c²

Also see

Proof of Cosine Rule

admin — Sun, 17 Feb 2008 16:31:46 +0000

Below is the proof by Pythagoras theorem of the cosine rule, a²=b²+c²- 2bccosA.

This assumes you understand Pythagoras theorem (visit Pythagoras theorem to view my lesson on it), how to use basic trigonometry(basic trigonometry lesson). If you want to learn how to use the cosine and sine rule, opposed to just learning the proof) visit by sine and cosine rule page.

The proof is done using the letters of the following triangle

and we are trying to prove the cosine rule:

a²=b²+c²- 2bccosA

In triangle CBL
a² = (c-x)² + h²
a² = c² – 2cx + x² + h²
h² = a² -c ²- x² + 2cx <

in triangle CLA
b² = h² + x²
h² = b² – x² <

eqn1 – eqn2 :: 0 = a² – c² – b² +2cx
a² = c ²+ b² – 2cx <

in CLA
cosA = x/b
x = bcosA

in eqn3

a² = c² + b² – 2bccosA

So there is the proof for the cosine rule using Pythagoras theorem.

Also see
Understand the Sine and Cosine Rules (11)
Trigonometry Identities (11)
Derive Quadratic Formula (16)
Proof of Pythagorases Theroem (0)
Area of a Triangle (3)
Proof by Contradiction (0)

Pythagorases Theorm

admin — Thu, 24 Jan 2008 21:30:00 +0000

The Theorem

“The square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides in a right angled triangle”

or mathematically

c² = b² + a²

This is very useful for allot of applications.

Applications of the theorem

There are many applications of Pythagoras besides simple triangles

eg)The difference between colors. Yes using pythag we can measure the “difference” between to colors as a number. This is how

take you first and second color as a rgb number eg) red = 256 ,0,0 and a light shade of blue is 30, 100, 256.

square the difference between the values for red, green and blue

then find the square root of the sum of these values

Why does this work, because the hypotenuse of one triangle can be used as the base of another, and this triangle can be tilted by 90 degrees into the z plane the theorem can work in 3D . We can then change the x,y and z axis to values for r,g,b. As pythag works in 3D we can calculate the distance between to colors(which are points in the rgb axis)

So there you go – Pythagoras can be used for measuring colors

Also see
Trigonometry Identities (11)
Proof of Pythagorases Theroem (0)
Area And Circumference of a Circle : pi (38)
Proof of Cosine Rule (4)