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This document provides a summary of key concepts in algebra, geometry, trigonometry and their definitions. It includes formulas and properties for lines, polynomials, exponents, trig functions, triangles, circles, spheres, cones, cylinders, distance and the quadratic formula. Key topics covered are factoring, binomials, slope-intercept form, trig ratios, trig identities, trig reciprocals and the Pythagorean identities.
![Trigonometry
(x, y) sin θ =
y Half-Angle
r
1 − cos 2θ 1 + cos 2θ
r sin2 θ = cos2 θ =
x 2 2
cos θ =
r
y
Addition
tan θ =
x sin(a + b) = sin a cos b + cos a sin b cos(a + b) = cos a cos b − sin a sin b
Subtraction
sin θ =
opp sin(a − b) = sin a cos b − cos a sin b cos(a − b) = cos a cos b + sin a sin b
hyp
hyp adj Sum
opp cos θ =
hyp u+v u−v
sin u + sin v = 2 sin cos
2 2
opp u+v u−v
tan θ = cos u + cos v = 2 cos cos
adj 2 2
adj
Product
sin u sin v = 1 [cos(u − v) − cos(u + v)]
2
Reciprocals cos u cos v = 1 [cos(u − v) + cos(u + v)]
2
1 1 1 sin u cos v = 1 [sin(u + v) + sin(u − v)]
2
cot θ = sec θ = csc θ =
tan θ cos θ sin θ cos u sin v = 1 [sin(u + v) − sin(u − v)]
2
Definitions π/2
2π/3 π/3
cos θ 1 1 π/4
cot θ = sec θ = csc θ = 3π/4
sin θ cos θ sin θ
5π/6 π/6
Pythagorean π 0
sin2 θ + cos2 θ = 1 tan2 θ + 1 = sec2 θ 1 + cot2 θ = csc2 θ Radians
sin(0) = 0 cos(0) = 1
√
π π
Cofunction sin 6 = 1
2 cos 6 = 2
3
√ √
π π
π π π sin = 2
cos = 2
sin 2 − θ = cos θ cos 2 − θ = sin θ tan 2 − θ = cot θ 4 2 4 2
√
π π
sin 3 = 2
3
cos 3 = 1
2
π π
sin =1 cos =0
Even/Odd 2
√
2
sin 2π
3 = 2
3
cos 2π
3 = −1
2
sin(−θ ) = −sin θ cos(−θ) = cos θ tan(−θ) = −tan θ √ √
sin 3π
4 = 2
2
cos 3π
4 =− 2
2
√
sin 5π
6 = 1
2 cos 5π
6 = − 23
Double-Angle sin(π) = 0 cos(π ) = −1
sin 2θ = 2 sin θ cos θ cos 2θ = cos2 θ − sin2 θ cos 2θ = 1 − 2 sin2 θ sin(2π) = 0 cos(2π) = 1](https://image.slidesharecdn.com/figures-111130202349-phpapp01/85/Figures-2-320.jpg)
Figures
- 1. Algebra Geometry Arithmetic Triangle Circle a+b a b a c ad + bc Area = 1 bh Area = πr 2 = + + = 2 c c c b d bd c2 = a 2 + b2 − 2ab cos θ C = 2πr a b a d ad = = a c b c bc c r h d b Factoring x 2 − y 2 = (x − y)(x + y) x 3 − y 3 = (x − y)(x 2 + x y + y 2 ) Sector of a Circle Trapezoid x 3 + y 3 = (x + y)(x 2 − x y + y 2 ) x 4 − y 4 = (x − y)(x + y)(x 2 + y 2 ) Area = 1 r 2 θ 2 Area = 1 (a + b)h 2 s = rθ Binomial (for θ in radians only) a (x + y)2 = x 2 + 2x y + y 2 (x + y)3 = x 3 + 3x 2 y + 3x y 2 + y 3 h s Exponents b xn xn xm = x n+m = x n−m (x n )m = x nm xm r n 1 x xn x −n = n (x y)n = x n y n = n x y y Sphere Cone √ √ √ √ √ x x n x n/m = m xn n xy = n xny n = √ Volume = 4 πr 3 Volume = 1 πr 2 h y n y 3 3 √ Surface Area = 4πr 2 Surface Area = πr r 2 + h 2 Lines Slope m of line through (x0 , y0 ) and (x1 , y1 ) r h y1 − y0 m= x1 − x0 r Through (x0 , y0 ), slope m y − y0 = m(x − x0 ) Slope m, y-intercept b Cylinder y = mx + b Volume = πr 2 h Surface Area = 2πr h Quadratic Formula If ax 2 + bx + c = 0 then √ −b ± b2 − 4ac h x= 2a r Distance Distance d between (x1 , y1 ) and (x2 , y2 ) d= (x2 − x1 )2 + (y2 − y1 )2
- 2. Trigonometry (x, y) sin θ = y Half-Angle r 1 − cos 2θ 1 + cos 2θ r sin2 θ = cos2 θ = x 2 2 cos θ = r y Addition tan θ = x sin(a + b) = sin a cos b + cos a sin b cos(a + b) = cos a cos b − sin a sin b Subtraction sin θ = opp sin(a − b) = sin a cos b − cos a sin b cos(a − b) = cos a cos b + sin a sin b hyp hyp adj Sum opp cos θ = hyp u+v u−v sin u + sin v = 2 sin cos 2 2 opp u+v u−v tan θ = cos u + cos v = 2 cos cos adj 2 2 adj Product sin u sin v = 1 [cos(u − v) − cos(u + v)] 2 Reciprocals cos u cos v = 1 [cos(u − v) + cos(u + v)] 2 1 1 1 sin u cos v = 1 [sin(u + v) + sin(u − v)] 2 cot θ = sec θ = csc θ = tan θ cos θ sin θ cos u sin v = 1 [sin(u + v) − sin(u − v)] 2 Definitions π/2 2π/3 π/3 cos θ 1 1 π/4 cot θ = sec θ = csc θ = 3π/4 sin θ cos θ sin θ 5π/6 π/6 Pythagorean π 0 sin2 θ + cos2 θ = 1 tan2 θ + 1 = sec2 θ 1 + cot2 θ = csc2 θ Radians sin(0) = 0 cos(0) = 1 √ π π Cofunction sin 6 = 1 2 cos 6 = 2 3 √ √ π π π π π sin = 2 cos = 2 sin 2 − θ = cos θ cos 2 − θ = sin θ tan 2 − θ = cot θ 4 2 4 2 √ π π sin 3 = 2 3 cos 3 = 1 2 π π sin =1 cos =0 Even/Odd 2 √ 2 sin 2π 3 = 2 3 cos 2π 3 = −1 2 sin(−θ ) = −sin θ cos(−θ) = cos θ tan(−θ) = −tan θ √ √ sin 3π 4 = 2 2 cos 3π 4 =− 2 2 √ sin 5π 6 = 1 2 cos 5π 6 = − 23 Double-Angle sin(π) = 0 cos(π ) = −1 sin 2θ = 2 sin θ cos θ cos 2θ = cos2 θ − sin2 θ cos 2θ = 1 − 2 sin2 θ sin(2π) = 0 cos(2π) = 1