What are these things called:
-cos x = cos -x
What are the rules to these trig rules?
Thanks
They are called identities. One of the characteristics of identities is that they apply to all angles x.
The above is not a correct identity, since correctly written, it would be:
cos(x) = cos(-x)
Other identities are
sin²(x)+cos²(x)=1
sin(2A)=2sin(A)cos(A)
...
Thanks for the prompt reply. How do these work out? The Odd/Even Identities?
sin -x = -sin x
cos -x = cox x
tan -x = -tanx
Why does this work out? Does it apply to certain quadrants or something?
Thanks.
The equation you mentioned "-cos x = cos -x" is an example of a trigonometric identity, specifically one involving the cosine function.
There are several trigonometric identities that involve the cosine function. The equation you provided is an example of the evenness property of cosine. The evenness property states that the cosine function is an even function, meaning that it is symmetric about the y-axis. This property allows us to conclude that:
cos x = cos -x
In other words, the cosine of an angle x is equal to the cosine of the negative of that angle, or equivalently, the cosine function is symmetric.
Trigonometric identities are important in solving trigonometric equations, simplifying expressions, and proving other mathematical statements involving trigonometric functions.
To understand and apply these trigonometric rules, it's helpful to have a solid foundation in basic trigonometry. Some of the commonly used trigonometric identities include:
1. Reciprocal identity:
- sin θ = 1/csc θ
- cos θ = 1/sec θ
- tan θ = 1/cot θ
2. Quotient identity:
- tan θ = sin θ / cos θ
- cot θ = cos θ / sin θ
3. Pythagorean identity:
- sin² θ + cos² θ = 1
- 1 + tan² θ = sec² θ
- 1 + cot² θ = csc² θ
4. Co-function identities:
- sin (π/2 - θ) = cos θ
- cos (π/2 - θ) = sin θ
- tan (π/2 - θ) = cot θ
- sec (π/2 - θ) = csc θ
- csc (π/2 - θ) = sec θ
- cot (π/2 - θ) = tan θ
These are just a few common examples; there are many more trigonometric identities that can be derived from these fundamentals.
To derive or prove trigonometric identities, you generally use algebraic manipulations or utilize geometric properties of triangles and circles. Understanding the basic properties and relationships between trigonometric functions is key to effectively using and applying these identities.