Trigonometric Identities

Main Trigonometric Identities

• sin²(x) + cos²(x) = 1

• sin(x+y) = sin(x)cos(y) + sin(y)cos(x)

• cos(x+y) = cos(x)cos(y) - sin(x)sin(y)

Tips 2

• If top and bottom are zero, factor and cancel.
• If degrees match, divide the leading coefficients.
• Smaller degree on top -> Add in an 0x => 0
• Trig limits, split up or multiply by common 1/1 to get sinx/x.
• For finding where a function is continuous, find where x makes parts undefined.
• Finding derivative, multiply by the conjugate, cancel h's so bottom is not zero.
• Product rule: (first)'(second) + (first)(second)'
• Quotient rule: (top)'(bottom) - (top)(bottom)' / (bottom)²
• Finding tangent line: y2 - y1 = f'(x)(x2 - x1)
• Don't forget the chain rule!
• Implicit differentiation, take derivative of both sides. Solve for y'.
• Taking derivative with respect to time gives you a function where you can plug in rates.
• Mean value theorem - f'(x) = (f(b) - f(a)) / (b - a)
• Linear approximation - Solve for y in tangent-line equation. y becomes L(x).
• Write dy in terms of x if y=... Set dy = f'(x) * dx.

Tips

• sin²x + cos²x = 1
• sin²x = (1-cos2x)/2
• For square roots, multiply by the conjugate
• For functions with a sign between them over a common denominator, multiply by the conjugate.
• Cross multiply
• Take the reciprocal and flip the inequality.
• Multiply/divide by a negative # and flip the inequality.
• Set the two functions equal to each other.
• Find where x != 0 in both numerator and denominator.
• Find the roots / zeroes of the function.
• Interval = range of x values (continuous).
• Plug in the derivative value.
• Plug in h = 0 when h is no longer a problem.

Discontinuities

• Removable
• Infinite
• Jump

Finding Limits

• Plug the limit value in.
• Factor and cancel
• Split limits apart and multiply to get sinx/x = 1 = x/sinx = (1-cosx)/x

Finding Limits at Infinity

• Factor our the highest power of x.

Proofs

• Needs |f(x) - limit value| < ε to |x - approaching| <> δ
• For infinite limit, needs |f(x)| > M to |x - approaching| <> M
• For limit at infinity, needs |f(x) - limit value| < ε to |x| <> δ