Essential Trigonometric Identities and Formulas

Written at March 6, 2025 on English with a size of 4.8 KB.

Pythagorean Identities: sin (a + b) = sin(a) · cos(b) + cos(a) · sin(b) cos (a + b) = cos(a) · cos(b) - sin(a) · sin(b) tan (a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b)) sin(2a) = 2 · sin(a) · cos(a) cos(2a) = cos²(a) - sin²(a) tan(2a) = 2tan(a) / (1 - tan²(a)) sin(a / 2) = ±√((1 - cos(a)) / 2) cos(a / 2) = ±√((1 + cos(a)) / 2) tan(a / 2) = ±√((1 - cos(a)) / (1 + cos(a))) sin(a)sin(b) = 2sin((a + b) / 2) · cos((a - b) / 2) sin(a) - sin(b) = 2cos((a + b) / 2) · sin((a - b) / 2) cos(a) + cos(b) = 2cos((a + b) / 2) · cos((a - b) / 2) cos(a) - cos(b) = -2sin((a + b) / 2) · sin((a - b) / 2)	Basic Trigonometric Identities: sin²(x) + cos²(x) = 1 1 + tan²(x) = sec²(x) 1 + cot²(x) = csc²(x) tan(x) = sin(x) / cos(x) csc(x) = 1 / sin(x) sec(x) = 1 / cos(x) cot(x) = 1 / tan(x) = cos(x) / sin(x) 1 + cot²(a) = csc²(a) sin (a + b) = sin(a) · cos(b) + cos(a) · sin(b) cos (a + b) = cos(a) · cos(b) - sin(a) · sin(b) sin (a - b) = sin(a) · cos(b) - cos(a) · sin(b) cos (a - b) = cos(a) · cos(b) + sin(a) · sin(b) sin(2a) = 2sin(a) · cos(a) cos(2a) = cos²(a) - sin²(a) tan(2a) = 2tan(a) / (1 - tan²(a)) sin (a / 2) = ±√((1 - cos(a)) / 2) cos (a / 2) = ±√((1 + cos(a)) / 2) tan (a / 2) = ±√((1 - cos(a)) / (1 + cos(a))) sin(a) + sin(b) = 2 · sin((a + b) / 2) · cos((a - b) / 2) sin(a) - sin(b) = 2 · cos((a + b) / 2) · sin((a - b) / 2) cos(a) + cos(b) = 2 · cos((a + b) / 2) · cos((a - b) / 2) cos(a) - cos(b) = -2 · sin((a + b) / 2) · sin((a - b) / 2) cos(a) = adjacent / hypotenuse => sec(a) sin(a) = opposite / hypotenuse => csc(a) tan(a) = opposite / adjacent => cotangent(a) = sin(a) / cos(a)
sin'(x) = cos(x) cos'(x) = -sin(x) tan'(x) = 1 + tan²(x) sin'(f(x)) = cos(f(x)) · f'(x) cos'(f(x)) = -sin(f(x)) · f'(x) tan'(f(x)) = [1 + tan²(f(x))] · f'(x) arcsin'(x) = 1 / √(1 - x²) arccos'(x) = -1 / √(1 - x²) arctan'(x) = 1 / (1 + x²) arcsin'(f(x)) = f'(x) / √(1 - f(x)²) arccos'(f(x)) = -f'(x) / √(1 - f(x)²) arctan'(f(x)) = f'(x) / (1 + f(x)²) log'(x) = 1 / x log'(f(x)) = f'(x) / f(x) log_k'(x) = 1 / (x · ln(k)) log_k'(f(x)) = f'(x) / (f(x) · ln(k)) e^x' = e^x (e^f(x))' = e^f(x) · f'(x) (k^f(x))' = k^f(x) · ln(k) · f'(x) k^x' = k^x · ln(k) [f(x)^k]' = k · f(x)^k-1 · f'(x) xⁿ' = nx^n-1 Sum Rule: D[f(x) + g(x)] = f'(x) + g'(x) Constant Multiple Rule: D[kf(x)] = kf'(x) Product Rule: D[f(x) · g(x)] = f'(x) · g(x) + f(x) · g'(x) Quotient Rule: D[f(x) / g(x)] = (f'(x) · g(x) - f(x) · g'(x)) / [g(x)]² Chain Rule: D(f[g(x)]) = f'[g(x)] · g'(x) D(f(g[h(x)])) = f'(g[h(x)]) · g'(h(x)) · h'(x) Power Rule Trigonometric Derivatives: D(sin(x)) = cos(x) D[sin(f(x))] = f'(x) · cos(f(x)) D(cos(x)) = -sin(x) D[cos(f(x))] = -f'(x) · sin(f(x)) D(tan(x)) = 1 + tan²(x) D(tan(f(x))) = (1 + tan²(f(x))) · f'(x) Exponential Derivatives: D(e^x) = e^x D[e^f(x)] = e^f(x) · f'(x) D(a^x) = a^x · ln(a) D[a^f(x)] = a^f(x) · ln(a) · f'(x) Logarithmic Derivatives:

Pythagorean Identities:

sin (a + b) = sin(a) · cos(b) + cos(a) · sin(b)
cos (a + b) = cos(a) · cos(b) - sin(a) · sin(b)
tan (a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b))
sin(2a) = 2 · sin(a) · cos(a)
cos(2a) = cos²(a) - sin²(a)
tan(2a) = 2tan(a) / (1 - tan²(a))
sin(a / 2) = ±√((1 - cos(a)) / 2)
cos(a / 2) = ±√((1 + cos(a)) / 2)
tan(a / 2) = ±√((1 - cos(a)) / (1 + cos(a)))
sin(a)sin(b) = 2sin((a + b) / 2) · cos((a - b) / 2)
sin(a) - sin(b) = 2cos((a + b) / 2) · sin((a - b) / 2)
cos(a) + cos(b) = 2cos((a + b) / 2) · cos((a - b) / 2)
cos(a) - cos(b) = -2sin((a + b) / 2) · sin((a - b) / 2)

Basic Trigonometric Identities:
sin²(x) + cos²(x) = 1
1 + tan²(x) = sec²(x)
1 + cot²(x) = csc²(x)
tan(x) = sin(x) / cos(x)
csc(x) = 1 / sin(x)
sec(x) = 1 / cos(x)
cot(x) = 1 / tan(x) = cos(x) / sin(x)
1 + cot²(a) = csc²(a)
sin (a + b) = sin(a) · cos(b) + cos(a) · sin(b)
cos (a + b) = cos(a) · cos(b) - sin(a) · sin(b)
sin (a - b) = sin(a) · cos(b) - cos(a) · sin(b)
cos (a - b) = cos(a) · cos(b) + sin(a) · sin(b)
sin(2a) = 2sin(a) · cos(a)
cos(2a) = cos²(a) - sin²(a)
tan(2a) = 2tan(a) / (1 - tan²(a))
sin (a / 2) = ±√((1 - cos(a)) / 2)
cos (a / 2) = ±√((1 + cos(a)) / 2)
tan (a / 2) = ±√((1 - cos(a)) / (1 + cos(a)))
sin(a) + sin(b) = 2 · sin((a + b) / 2) · cos((a - b) / 2)
sin(a) - sin(b) = 2 · cos((a + b) / 2) · sin((a - b) / 2)
cos(a) + cos(b) = 2 · cos((a + b) / 2) · cos((a - b) / 2)
cos(a) - cos(b) = -2 · sin((a + b) / 2) · sin((a - b) / 2)
cos(a) = adjacent / hypotenuse => sec(a)
sin(a) = opposite / hypotenuse => csc(a)
tan(a) = opposite / adjacent => cotangent(a) = sin(a) / cos(a)

sin'(x) = cos(x)
cos'(x) = -sin(x)
tan'(x) = 1 + tan²(x)
sin'(f(x)) = cos(f(x)) · f'(x)
cos'(f(x)) = -sin(f(x)) · f'(x)
tan'(f(x)) = [1 + tan²(f(x))] · f'(x)
arcsin'(x) = 1 / √(1 - x²)
arccos'(x) = -1 / √(1 - x²)
arctan'(x) = 1 / (1 + x²)
arcsin'(f(x)) = f'(x) / √(1 - f(x)²)
arccos'(f(x)) = -f'(x) / √(1 - f(x)²)
arctan'(f(x)) = f'(x) / (1 + f(x)²)

log'(x) = 1 / x
log'(f(x)) = f'(x) / f(x)
log_k'(x) = 1 / (x · ln(k))
log_k'(f(x)) = f'(x) / (f(x) · ln(k))

e^x' = e^x
(e^f(x))' = e^f(x) · f'(x)
(k^f(x))' = k^f(x) · ln(k) · f'(x)
k^x' = k^x · ln(k)
[f(x)^k]' = k · f(x)^k-1 · f'(x)
xⁿ' = nx^n-1
Sum Rule:
D[f(x) + g(x)] = f'(x) + g'(x)
Constant Multiple Rule:
D[kf(x)] = kf'(x)
Product Rule:
D[f(x) · g(x)] = f'(x) · g(x) + f(x) · g'(x)
Quotient Rule:
D[f(x) / g(x)] = (f'(x) · g(x) - f(x) · g'(x)) / [g(x)]²
Chain Rule:
D(f[g(x)]) = f'[g(x)] · g'(x)
D(f(g[h(x)])) = f'(g[h(x)]) · g'(h(x)) · h'(x)
Power Rule
Trigonometric Derivatives:
D(sin(x)) = cos(x)
D[sin(f(x))] = f'(x) · cos(f(x))
D(cos(x)) = -sin(x)
D[cos(f(x))] = -f'(x) · sin(f(x))
D(tan(x)) = 1 + tan²(x)
D(tan(f(x))) = (1 + tan²(f(x))) · f'(x)
Exponential Derivatives:
D(e^x) = e^x
D[e^f(x)] = e^f(x) · f'(x)
D(a^x) = a^x · ln(a)
D[a^f(x)] = a^f(x) · ln(a) · f'(x)
Logarithmic Derivatives:

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