How can I memorize trigonometric identities without rote memorization?

Question

I have a trig final next week and Im drowning in identities.

Theres like 50 of them:

Pythagorean identities
Angle addition formulas
Double angle formulas
Half angle formulas
Sum-to-product
Product-to-sum
Power-reducing formulas

I cant memorize all of these! Is there a way to learn them by understanding the connections?

I know that most of them come from:

\cos(A+B) = \cos A \cos B - \sin A \sin B

\sin(A+B) = \sin A \cos B + \cos A \sin B

And using $A = B$ gives double angle formulas. Using $\cos^2 A + \sin^2 A = 1$ gives the other Pythagorean forms.

But the sum-to-product formulas are still hard. And the half-angle formulas with the $\pm$ signs confuse me.

Also: is it true that Euler formula $e^{i\theta} = \cos\theta + i\sin\theta$ can generate ALL trig identities automatically? How would that work?

Chen Wei · Accepted Answer

Heres a systematic approach that reduces memorization to just THREE core identities: Core identities (memorize these): 1. sin ⁡ 2 θ + cos ⁡ 2 θ = 1 2. sin ⁡ ( A + B ) = sin ⁡ A cos ⁡ B + cos ⁡ A sin ⁡ B 3. cos ⁡ ( A + B ) = cos ⁡ A cos ⁡ B − sin ⁡ A sin ⁡ B Everything else comes from these: Double angle: Set B = A: sin ⁡ 2 A = 2 sin ⁡ A cos ⁡ A cos ⁡ 2 A = cos ⁡ 2 A − sin ⁡ 2 A = 2 cos ⁡ 2 A − 1 = 1 − 2 sin ⁡ 2 A Half angle: Solve cos ⁡ 2 θ = 1 − 2 sin ⁡ 2 θ for sin ⁡ θ: sin ⁡ θ 2 = ± 1 − cos ⁡ θ 2 Using Eulers formula (e i θ = cos ⁡ θ + i sin ⁡ θ) is the most elegant approach: e i ( A + B ) = e i A e i B cos ⁡ ( A + B ) + i sin ⁡ ( A + B ) = ( cos ⁡ A + i sin ⁡ A ) ( cos ⁡ B + i sin ⁡ B ) Multiply out and equate real/imaginary parts — you get both addition formulas instantly! For product-to-sum: Use the addition formulas with P = ( A + B ) / 2 and Q = ( A − B ) / 2. The ± signs in half-angle formulas are determined by which quadrant θ / 2 lies in. On a unit circle, you can derive this visually.

Alex Kim · Answer

I would not start with a giant list. Start with a small family tree. The identities I would keep as "parents" are: sin ⁡ 2 x + cos ⁡ 2 x = 1 and the angle addition formulas: sin ⁡ ( a + b ) = sin ⁡ a cos ⁡ b + cos ⁡ a sin ⁡ b cos ⁡ ( a + b ) = cos ⁡ a cos ⁡ b − sin ⁡ a sin ⁡ b . From those, double-angle formulas come from setting a = b = x. Power-reducing formulas come from rearranging the double-angle formulas. Many sum-to-product formulas come from adding and subtracting angle formulas. So the study method is: 1. memorize the parent identities, 2. practice deriving the children, 3. keep one page showing the family tree, 4. only memorize the formulas you cannot derive quickly enough for your exam. That feels slower for one day, but much faster after a week because the identities stop looking unrelated.

How can I memorize trigonometric identities without rote memorization?

2 Answers

How can I memorize trigonometric identities without rote memorization?

2 Answers