Algebra and Trigonometry

# Key Terms

altitude
a perpendicular line from one vertex of a triangle to the opposite side, or in the case of an obtuse triangle, to the line containing the opposite side, forming two right triangles
ambiguous case
a scenario in which more than one triangle is a valid solution for a given oblique SSA triangle
Archimedes’ spiral
a polar curve given by$r=θ. r=θ.$When multiplied by a constant, the equation appears as$r=aθ. r=aθ.$As$r=θ, r=θ,$the curve continues to widen in a spiral path over the domain.
argument
the angle associated with a complex number; the angle between the line from the origin to the point and the positive real axis
cardioid
a member of the limaçon family of curves, named for its resemblance to a heart; its equation is given as$r=a±bcos θ r=a±bcos θ$ and$r=a±bsin θ, r=a±bsin θ,$where$a b =1 a b =1$
convex limaҫon
a type of one-loop limaçon represented by$r=a±bcos θ r=a±bcos θ$and$r=a±bsin θ r=a±bsin θ$such that$a b ≥2 a b ≥2$
De Moivre’s Theorem
formula used to find the$nth nth$power or nth roots of a complex number; states that, for a positive integer$n, z n n, z n$is found by raising the modulus to the$nth nth$power and multiplying the angles by$n n$
dimpled limaҫon
a type of one-loop limaçon represented by$r=a±bcos θ r=a±bcos θ$ and$r=a±bsin θ r=a±bsin θ$ such that$1< a b <2 1< a b <2$
dot product
given two vectors, the sum of the product of the horizontal components and the product of the vertical components
Generalized Pythagorean Theorem
an extension of the Law of Cosines; relates the sides of an oblique triangle and is used for SAS and SSS triangles
initial point
the origin of a vector
inner-loop limaçon
a polar curve similar to the cardioid, but with an inner loop; passes through the pole twice; represented by$r=a±bcos θ r=a±bcos θ$ andwhere$a
Law of Cosines
states that the square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of the other two sides and the cosine of the included angle
Law of Sines
states that the ratio of the measurement of one angle of a triangle to the length of its opposite side is equal to the remaining two ratios of angle measure to opposite side; any pair of proportions may be used to solve for a missing angle or side
lemniscate
a polar curve resembling a figure 8 and given by the equation$r 2 = a 2 cos 2θ r 2 = a 2 cos 2θ$and$r 2 = a 2 sin 2θ, a≠0 r 2 = a 2 sin 2θ, a≠0$
magnitude
the length of a vector; may represent a quantity such as speed, and is calculated using the Pythagorean Theorem
modulus
the absolute value of a complex number, or the distance from the origin to the point$( x,y ); ( x,y );$also called the amplitude
oblique triangle
any triangle that is not a right triangle
one-loop limaҫon
a polar curve represented by$r=a±bcos θ r=a±bcos θ$ and$r=a±bsin θ r=a±bsin θ$ such that $a>0,b>0, a>0,b>0,$and$a b >1; a b >1;$ may be dimpled or convex; does not pass through the pole
parameter
a variable, often representing time, upon which$x x$and$y y$are both dependent
polar axis
on the polar grid, the equivalent of the positive x-axis on the rectangular grid
polar coordinates
on the polar grid, the coordinates of a point labeled$( r,θ ), ( r,θ ),$where$θ θ$indicates the angle of rotation from the polar axis and$r r$represents the radius, or the distance of the point from the pole in the direction of$θ θ$
polar equation
an equation describing a curve on the polar grid.
polar form of a complex number
a complex number expressed in terms of an angle $θ θ$ and its distance from the origin$r; r;$can be found by using conversion formulas$x=rcos θ, y=rsin θ, x=rcos θ, y=rsin θ,$and$r= x 2 + y 2 r= x 2 + y 2$
pole
the origin of the polar grid
resultant
a vector that results from addition or subtraction of two vectors, or from scalar multiplication
rose curve
a polar equation resembling a flower, given by the equations$r=acos nθ r=acos nθ$ and $r=asin nθ; r=asin nθ;$when $n n$ is even there are $2n 2n$ petals, and the curve is highly symmetrical; when$n n$is odd there are $n n$ petals.
scalar
a quantity associated with magnitude but not direction; a constant
scalar multiplication
the product of a constant and each component of a vector
standard position
the placement of a vector with the initial point at$( 0,0 ) ( 0,0 )$and the terminal point$(a,b), (a,b),$represented by the change in the x-coordinates and the change in the y-coordinates of the original vector
terminal point
the end point of a vector, usually represented by an arrow indicating its direction
unit vector
a vector that begins at the origin and has magnitude of 1; the horizontal unit vector runs along the x-axis and is defined as$v 1 =〈 1,0 〉 v 1 =〈 1,0 〉$the vertical unit vector runs along the y-axis and is defined as$v 2 =〈 0,1 〉. v 2 =〈 0,1 〉.$
vector
a quantity associated with both magnitude and direction, represented as a directed line segment with a starting point (initial point) and an end point (terminal point)