Trig ientities
http://www.sosmath.com/trig/Trig5/trig5/trig5.html
Trig identites
(1/1+tanx)+(1/1+cotx)=1
For this one, I wanted to know if 1+tanx is the same as 1+tan^2x, because 1+tan^2x=sec^2x
These are the ones that I have no idea how to do:
((cosx-cosy)/(sinx+siny))+((sinx-siny)/(cosx+cosy))=0
(sin^3x+cos^3x)/(sinx+cosx)=1-sinxcosx
csc2x=cscx/2cosx
cos3x/cosx=1-4sin^2x
**sorry I didn't type it the first time, my computer froze and posted it like that**
no, 1+tanx is not the same as 1+tan^2x
use your sum-to-product formulas:
cosx-cosy = -2 sin(x+y)/2 sin(x-y)/2
sinx+siny = 2 sin(x+y)/2 cos(x-y)/2
so the quotient is just -tan(x-y)/2
the 2nd term is similar
recall that (a+b)^3 = (a+b)(a^2-ab+b^2)
csc2x = 1/sin2x = 1/(2sinx cosx)
cos3x = cos(2x+x)
= cos2x cosx - sin2x sinx
and so on
Trigonometric identities are mathematical equations that relate the values of trigonometric functions. These identities are valuable in solving trigonometric equations and simplifying expressions involving trigonometric functions.
There are various trigonometric identities, and it would be difficult to explain all of them comprehensively in a single response. However, I'll provide an overview of some essential trigonometric identities and explain how to derive them.
1. Pythagorean Identities:
- sin²θ + cos²θ = 1
- 1 + cot²θ = csc²θ
- tan²θ + 1 = sec²θ
To derive the Pythagorean identities, start with the unit circle or a right-angled triangle with sides labeled as a, b, and c (hypotenuse). Apply the definitions of sine, cosine, tangent, cotangent, secant, and cosecant to the triangle, and then use the Pythagorean theorem to relate the sides. Simplify the equations to obtain the Pythagorean identities.
2. Double-Angle Identities:
- sin(2θ) = 2sinθcosθ
- cos(2θ) = cos²θ - sin²θ
- tan(2θ) = (2tanθ) / (1 - tan²θ)
The double-angle identities can be derived by using the sum and difference identities for sine, cosine, and tangent and substituting θ = θ + θ.
3. Sum and Difference Identities:
- sin(α ± β) = sinαcosβ ± cosαsinβ
- cos(α ± β) = cosαcosβ ∓ sinαsinβ
- tan(α ± β) = (tanα ± tanβ) / (1 ∓ tanαtanβ)
The sum and difference identities can be derived by using the formulas for sine, cosine, and tangent of the sum of two angles and simplifying the expressions.
These are just a few of the many trigonometric identities. To explore additional identities, you may refer to trigonometry books, online resources, or trigonometry textbooks, which contain comprehensive tables of identities and their derivations.