Trig identities
Trigonometric identities are equations that are used to describe the many relationships that exist between the trigonometric functions. Among other uses, they can be helpful for simplifying trigonometric expressions and equations.
The following shows some of the identities you may encounter in your study of trigonometry.
Reciprocal identities
sin(θ)·csc(θ) = 1
cos(θ)·sec(θ) = 1
tan(θ)·cot(θ) = 1
Quotient identities
Cofunction identities
Odd/even identities
sin(-θ) = -sin(θ)
cos(-θ) = cos(θ)
tan(-θ) = -tan(θ)
csc(-θ) = -csc(θ)
sec(-θ) = sec(θ)
cot(-θ) = -cot(θ)
Pythagorean identities
cos2(θ) + sin2(θ) = 1
1 + tan2(θ) = sec2(θ)
1 + cot2(θ) = csc2(θ)
Example:
Verify that cos(x)·tan(x) + csc(x)·cos2(x) = csc(x) using trigonometric identities.
| cos(x)·tan(x) + csc(x)·cos2(x) | |
| = |  |
| = |  |
| = |  |
| = |  |
| = | csc(x) |
Trigonometric formulas
There are many formulas used in trigonometry that involve one or more angles or sides of a triangle.
Sum and difference formulas
| sin(x ± y) = sin(x)·cos(y) ± cos(x)·sin(y) |
| cos(x ± y) = cos(x)·cos(y) ∓ sin(x)·sin(y) |
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Double angle formulas
| sin(2θ) = 2·sin(θ)·cos(θ) |
| cos(2θ) = cos2(θ) - sin2(θ) = 1 - 2·sin2(θ) = 2·cos2(θ) - 1 |
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Half angle formulas
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Less frequently used identities
Though the identities below are not used as frequently as some of those above, you may still come across them in your studies.
Product to sum identities
| 2·sin(x)·cos(y) = sin(x+y) + sin(x-y) |
| 2·cos(x)·sin(y) = sin(x+y) - sin(x-y) |
| 2·cos(x)·cos(y) = cos(x+y) + cos(x-y) |
| 2·sin(x)·sin(y) = cos(x-y) - cos(x+y) |
Sum to product identities
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Example:
Find the exact value of cos(105°) + cos(15°) using a sum to product identity:
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