Mathematics: Formulas (ACADEC '25-'26)

Slope-Intercept Form

y = mx + b

Point-Slope Formula

y - y1 = m(x - x1)

Quadratic Form

ax² + bx + c = 0

What does ‘x’ equal in the form x² - p = 0?

x = ±√p

What does ‘x’ equal in the form k(x+r)² - p = 0, where k ≠ 0?

x = -r ± √(p/k)

When does the quadratic in the form k(x+r)² - p = 0 have no real solution?

If p/k < 0

Quadratic Formula

What is the formula for the discriminant?

b² - 4ac

What’re the rules for the discriminant?

1. When the discriminant > 0, two real solutions

2. When the discriminant = 0, one real solution

3. When the discriminant < 0, no real solution

Polynomial Form

Monomial Form

x^k

Complex Form

a + bi

Conjugate of Complex Form

a - bi

What is a complex form multiplied by its conjugate equal to?

a² + b²

Where does a linear graph intersect the x-axis when a ≠ 0?

x = -b/a

What is ‘k’ equivalent to in quadratic form?

What can quadratic form ax² + bx + c be shifted to be equivalent to?

What are the three properties of ‘a’ in relation to the min/max of a quadratic equation?

1. When a > 0, the graph has a minimum at x = -b/2a

2. When a < 0, the graph has a maximum at x = -b/2a

3. The graph is symmetric with respect to the line x = -b/2a

Distance Formula

Exponential Form

y = a^x

Logarithmic Form

log_b(a) = c

Natural Logarithmic Form

log_e(b) = c

Reciprocal Function

y = 1/x

Change of Base Formula

What is the less compact form of |x|?

Pythagorean Theorem

a² + b² = c²

what are the six basic relations among trigonometric functions?

1. tanα = sinα/cosα

2. cotα = cosα/sinα

3. tanα * cotα = 1

4. sin²α + cos²α = 1

5. tan2α + 1 = α = 1/cos²α, cosα ≠ 0

6. 1 + cot²α = 1/sin²α, sinα ≠ 0

what can the identity “tan2α + 1 = α = 1/cos²α, cosα ≠ 0?” be rewritten as?

tan²α + 1 = sec²α

what can the identity “1 + cot²α = 1/sin²α, sinα ≠ 0” be rewritten as?

1 + cot²α = csc²α

what are the 4 negative angle identites?

1. sin(-α) = -sinα

2. cos(-α) = cosα

3. tan(-α) = -tanα

4. cot(-α) = -cotα

what are the 4 sum & difference identities?

1. sin(α + B) = sinαcosB + cosαsinB

2. cos(α + B) = cosαcosB - sinαsinB

3. sin(α - B) = sinαcosB - cosαsinB

4. cos(α - B) = cosαcosB + sinαsinB

what is the sine double-angle identity?

sin2α = 2sinαcosα

what is the cosine double-angle identity?

cos2α = cos²α - sin²α

what is the tangent double-angle identity?

tan2α = 2tanα/(1-tan²α)

what is the sine half-angle identity?

what is the cosine half-angle identity?

what is the tangent half-angle identity?

what is the cotangent half-angle identity?

what are the two sine sum-to-product identities?

what are the two cosine sum-to-product identities?

what are the two tangent sum-to-product identities?

what are the two cotangent sum-to-product identities?

what is the sine product-to-sum identity?

what is the cosine product-to-sum identity?

what is sine + cosine product-to-sum identity?

what is the law of sines?

what is the law of cosines?

what is vieta’s formula?

what’s the equation for a circle?

(x - h)² + (y - k)² = r²

what is midpoint formula?

in a 30-60-90 triangle, what are the rules of the sides?

in a 45-45-90 triangle, what are the rules of the sides?

what’s the formula to convert a degree measure to a radian measure?

multiply the degree measure by the conversion factor π/180

what is 1/sinα often denoted by?

cscα

what is 1/cosα often denoted by?

secα

what is cos(90 degrees - α) equal to?

sinα

what is sin(90 degrees - α) equal to?

cosα

what is tan(90 degrees - α) equal to?

cotα

what is cot(90 degrees - α) equal to?

tanα

what is sec(90 degrees - α) equal to?

cscα

what is csc(90 degrees - α) equal to?

secα

the expression ‘i’ is equivalent to?

sqrt(-1)

what’s the formula for the angle theta between two lines with slope m1 & m2?