Difference between revisions of "Trigonometric identities"
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* <math>\cot (-x) = -\cot (x) </math> | * <math>\cot (-x) = -\cot (x) </math> | ||
− | These are derived by the unit circle definitions of trigonometry. Making any angle negative is the same as | + | These are derived by the unit circle definitions of trigonometry. Making any angle negative is the same as reflecting it across the x-axis. This keeps its x-coordinate the same, but makes its y-coordinate negative. Thus, <math>\sin(-x) = -\sin(x)</math> and <math>\cos(-x) = \cos(x)</math>. |
== Pythagorean Identities == | == Pythagorean Identities == |
Revision as of 18:10, 30 May 2021
Trigonometric Identities are used to manipulate trigonometry equations in certain ways. Here is a list of them:
Contents
Even-Odd Identities
The functions ,
, and
are odd, while
,
, and
are even. In other words, the six trigonometric functions satisfy the following equalities:
These are derived by the unit circle definitions of trigonometry. Making any angle negative is the same as reflecting it across the x-axis. This keeps its x-coordinate the same, but makes its y-coordinate negative. Thus, and
.
Pythagorean Identities
The Pythagorean identities state that
Using the ratio definition of trigonometry, we apply Pythagorean Theorem on our triangle above to get that . If we divide by
, we get
, which is just
. Using the unit circle definition of trigonometry, it's extremely easy to see that
.
To derive the other two Pythagorean identities, divide by either or
and substitute the respective trigonometry in place of the ratios.
Angle Addition Identities
The trigonometric angle addition identities state the following identities:
There are many proofs of these identities. For the sake of brevity, we list only one here.
Euler's identity states that . We have that
By looking at the real and imaginary parts, we derive the sine and cosine angle addition formulas.
To derive the tangent addition formula, we reduce tangent to sine and cosine, divide both numerator and denominator by , and simplify.
as desired.
Double-Angle Identities
The trigonometric double-angle identities are easily derived from the angle addition formulas by just letting . Doing so yields:
Cosine Double-Angle
Here are two equally useful forms of the cosine double-angle identity. Both are derived via the Pythagorean identity on the cosine double-angle identity given above.
In addition, the following identities are useful in integration and in deriving the half-angle identities. They are a simple rearrangement of the two above.
Half-Angle Identities
The trigonometric half-angle identities state the following equalities:
The plus or minus is not saying that there are two answers, but that the sine of the angle depends on the quadrant that the angle is in.
Take the two expressions listed in the cosine double-angle section for and
, and substitute
instead of
. Taking the square root then yields the desired half-angle identities for sine and cosine.
Sum-to-Product Identities
Law of Sines
- Main article: Law of Sines
The extended Law of Sines states
Law of Cosines
- Main article: Law of Cosines
The Law of Cosines states
Law of Tangents
- Main article: Law of Tangents
The Law of Tangents states that if and
are angles in a triangle opposite sides
and
respectively, then
A further extension of the Law of Tangents states that if ,
, and
are angles in a triangle, then
Other Identities
(This is also written as
)
The two identities above are derived from the Pythagorean Identities.