Hi !
Here is my question:
The pythagorean theorem gives the relationships between the sides of a right triangle. The following identities show the relationships between the trigonometric functions of a particular angle.
sin^2(theta) + cos^2(theta) = 1
tan^2(theta) + 1 = sec^2(theta)
1 + cot^2(theta) = csc^2(theta)
Why are these referred to the pythagorean identities?
Make a sketch of a right-angle triangle with hypotenuse of 1, base side (the horizontal) x, and vertical side y , call the base angle Ø
by Pythagoras:
x^2 + y^2 = 1^2 = 1
isn't cosØ = x/1 = x
then x^2 = cos^2
and isn't sinØ = y/1 = 1
then y^2 = sin^2 Ø
so in x^2 + y^2 = 1
sin^2 Ø + cos^2 Ø = 1
so we proven this important relationship and since it was base on the Pythagorean Theorem we call it a Pythagorean identity.
the other two are obtained in a very simple way
for the 1st, divide each term by cos^2 Ø
sin^2 Ø/cos^2 Ø + cos^2 Ø = 1/cos^2 Ø
tan^2 Ø + 1 = sec^2 Ø
the last is obtained by dividing each term of the original by sin^2 Ø
You try it.
Thank you sir
The trigonometric identities you mentioned are referred to as the Pythagorean identities because they closely resemble the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Similarly, the Pythagorean identities show relationships between the trigonometric functions (sine, cosine, tangent, secant, cosecant, and cotangent) for a given angle theta. These identities are derived from the properties of right triangles and the ratios between the sides of those triangles.
Let's take a closer look at each identity:
1. sin^2(theta) + cos^2(theta) = 1:
This identity represents the relationship between the sine and cosine of an angle. For any angle theta, if you square the value of the sine and add it to the square of the cosine, the sum will always equal 1. This is similar to the Pythagorean theorem, as it shows the added squares of two values resulting in a constant value (1).
2. tan^2(theta) + 1 = sec^2(theta):
This identity relates the tangent and secant functions of an angle. It states that if you square the value of the tangent and add 1 to it, the sum will always be equal to the square of the secant. Again, this resembles the Pythagorean theorem because it shows the sum of the square of one function and a constant value (1) resulting in the square of another function.
3. 1 + cot^2(theta) = csc^2(theta):
This identity shows the relationship between the cotangent and cosecant functions of an angle. According to this identity, if you add 1 to the square of the cotangent, the result will always be equal to the square of the cosecant. Once again, this is similar to the Pythagorean theorem as it demonstrates the sum of 1 and the square of one function resulting in the square of another function.
Overall, the Pythagorean identities are named after the Pythagorean theorem because they exhibit similar mathematical patterns and relationships, involving the squares of trigonometric functions instead of the squares of triangle side lengths. These important identities are widely used in trigonometry and have various applications in mathematics and science.