Trig Identities
Trigonometric identities are equations that are true for all values of the variables involved, where the variables typically represent angles
Trigonometric identities are equations that are true for all values of the variables involved, where the variables typically represent angles. These identities are essential in simplifying trigonometric expressions, solving equations, and proving other mathematical theorems.
Here are some of the most commonly used trigonometric identities:
1. Pythagorean identities:
– sin^2(x) + cos^2(x) = 1
– tan^2(x) + 1 = sec^2(x)
– 1 + cot^2(x) = csc^2(x)
2. Reciprocal identities:
– sin(x) = 1/csc(x)
– cos(x) = 1/sec(x)
– tan(x) = 1/cot(x)
3. Quotient identities:
– tan(x) = sin(x)/cos(x)
– cot(x) = cos(x)/sin(x)
4. Co-Function identities:
– sin(π/2 – x) = cos(x)
– cos(π/2 – x) = sin(x)
– tan(π/2 – x) = cot(x)
– cot(π/2 – x) = tan(x)
– sec(π/2 – x) = csc(x)
– csc(π/2 – x) = sec(x)
5. Angle sum and difference identities:
– sin(x ± y) = sin(x)cos(y) ± cos(x)sin(y)
– cos(x ± y) = cos(x)cos(y) ∓ sin(x)sin(y)
– tan(x ± y) = (tan(x) ± tan(y))/(1 ∓ tan(x)tan(y))
6. Double angle identities:
– sin(2x) = 2sin(x)cos(x)
– cos(2x) = cos^2(x) – sin^2(x)
– tan(2x) = 2tan(x)/(1 – tan^2(x))
7. Half angle identities:
– sin(x/2) = ±√[(1 – cos(x))/2]
– cos(x/2) = ±√[(1 + cos(x))/2]
– tan(x/2) = ±√[(1 – cos(x))/(1 + cos(x))]
These are just a few of the fundamental trigonometric identities, and there are many more that can be derived by manipulating and combining these basic identities. It is crucial to understand these identities thoroughly to solve trigonometric problems effectively.
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