ALL MATH

Find the midpoint between (-3, -8) and (1, -2).

(-1, -5)

What is the range of P(t) =-3t² - 6t +5?

(-∞,8)

Find the midpoint between the points (6, -5) and (-6, 7).

(0,1)

Solve: (7 - 2i) / (3 - i)

(23 + i) / 10

Solve: (3 + 6i) / (3 + 5i)

(39 + 3i) / 34

solve x×(y+z)

(X×Y) + (X×Z)

what is the equation of a circle?

(x-m)² +(y-n)² = r² with radius r centered at C(m, n)

For complex numbers what does i^2 equal?

-1

sin (270°)

-1

solve: cos (135°)

-1/√2

What is the domain of y=arccos x?

-1≤x≤1

State the sum to identity for cosa - cosb

-2 sin a+b/2 sin a-b/2

Add the following Polynomials: Q(x)=7x⁴−3x³+2x²−5x+4=0 P(x)=-2x⁵−2x⁴+3x³−4x²+10x+6

-2x⁵+5x⁴+0x³−2x²+5x+10

Multiply: P(x)=5x+5 Q(x)=-6x²−3x

-30x³−45x²−15x

Solve arcsin (sin(3π/4)

-3π/4

Given that sin a =4/5, solve tan 2a -2sin a/ 1-tan² a

-48/35

What is the minimum value of y = -4 cos (x/2 +π/6) -2

-6

Solve (1/2 + 3/4 i) * (3/2 + 4i)

-9/4 + 25/8 i

What is arcsin (-1) in degrees?

-90⁰ or 270 degrees

Which values of x satisfy the equation arctan (tanx) =x?

-90⁰<x<90⁰

What is the range of y= arc sin x?

-90⁰≤y≤90⁰

Solve: cot (-α)

-cot α

solve: sin (-a)

-sin a

Simplify the expression cos (-x) tan (-x)

-sinx

Solve: tan (-α)

-tan α

If x is located to the left of 0, then the absolute value of x is

-x

What is the arcsin(-1) in radians?

-π/2 or 3π/2

Using principal values, find arcsin(-1/2)

-π/6

Find: sec 135°

-√2

Sin (585⁰) is equal to

-√2/2

What values satisfy ax² + bx + c > 0, where ∆ = b² -4ac < 0 and a > 0?

-∞ < x < ∞

At 0° , y=0, what is the sin (0°)?

0

Solve tan (720°)

0

What is the arcos(1) in radians?

0

What is the arcsin(0) in radians?

0

What radius of a circle, will not satisfy the radians angle ratio?

0

solve: cos90°? sin90°

0

solve: sin (180°)

0

What condition must an acute angle, a, satisfy?

0°< a < 90°

What is the arcsin(0) in degrees?

0⁰, 180⁰

Which values of x satisfies arccot (cotx) =x?

0⁰<x<180⁰

Which values of x satisfy the equation arccos (cosx)= x?

0⁰≤x≤180⁰

Solve cos[arcos (3/5) - arcsin (4/5)]

1

What is sin (90°), where y=r?

1

solve: cos (0°)

1

solve: cos (360°)

1

solve: sin² a + cos² a

1

solve: tan a * cot a

1

solve:cot 45°

1

tan (x) cos (x) csc (x) is equivalent to

1

State the cot a/2 half angle identity

1+cos a/sin a

In radians, tan−¹ (2.2) is closest to

1.14

For an acute angle, the cosecant value is 10. Find the sine value of this angle.

1/10

solve cot60°*sin60°

1/2

solve: cos°

1/2

State the product to sum identity for cos a cos b

1/2 (cos (a+b) + cos (a-b))

State the product to sum identity for sin a sin b

1/2 (cos (a-b) -cos (a+b))

State the product to sum identity for sin a cos b.

1/2 (sin(a+b) + sin (a-b))

What is the formula for the area of a triangle?

1/2 AB*AC*sin(a)

The graph of the function y=(ax) shrinks or expands along the x-axis by what factor?

1/a

What ratio is the radians angle a defined to be for a circle with radius r and the length l of the arc intercepted by alpha?

1/r

What is the amplitude of y=-10 cos(2x-5)?

10

What is the absolute value of -1150?

1150

Find the angle measure of 2π/3

120⁰

Find the angle measure of π.

180⁰

What is the period of a cotangent function?

180⁰

What is the period of a tangent function?

180⁰

tan² a +1=?

1? cos²; cos a≠0

Evaluate tan 45⁰/cos 60⁰

2

Simplify the expression (cos x +sin x)² + ( cos x-sin x)²

2

What is the radian measure of the angle with an arc length of 4 and radius 2?

2

When the discriminant of a quadratic equation equals b²−4ac>0, how many real solutions does it have?

2

Solve tan(2a)

2 tan a/1-(tan)^2 a

Find tan 165⁰

2+√3

If Sally left at 8:30 am central time and arrived at 12:00 pm eastern, how long did Sally's trip take?

2.5 hours

Solve: (7 +3i) + (6+9i) + (8+6i)

21+18i

Find the angle measure of 7π/6

210⁰

Given that sin a =3/5 where 0<a<90⁰, find sin 2a.

24/25

State the sum to product identity for cosa + cosb

2cos a+b/2 cos a-b?2

Subtract the following polynomials (Q(x)−R(x)): Q(x)=x⁶−4x²+3x−2 R(x)=-x⁶+4x²−3x+2

2x⁶−8x²+6x−4

What is the remainder if x⁴+2x³+2x²−7 is divided by x²+2x+3

2x−4

Find the radian measure of 360⁰.

2π

What is the radian measure of a, where the degree of the angle a is 360⁰?

2π

solve: cot 30°/ sin30°

2√3

Given that sin a =4/5, solve cot a

3/4

Sara multiplied a number by 10 and added 15 to the results. Her total was 315. What was Sara's intial number?

30

Find the angle measure of 5π/3

300⁰

What is the arcsin (1/2) in degrees?

30⁰, 150⁰

Find the angle measure of 7π/4

315⁰

Steve divided a number by 6, then multiplied it by 2 and added 26. His total was 38. What was Steven's initial number?

36

What is the period of a cosine function?

360⁰

What is the period of a sine function?

360⁰

Find the radian measure of 270⁰.

3π/2

Solve arc cos (-1) + arc cos0, in radians

3π/2

Solve arccot (1) +arc sin 1, in radians

3π/4

Find the angle measure of π/4.

45⁰

What is the arcsine of √2/2?

45⁰

Subtract the following polynomials (Q(x)−R(x)): Q(x)=4x⁴−x²+6 R(x)=3x³+x−6

4x⁴−3x³−x²−x+12

What is the amplitude of y=10-5 cos(3x)?

5

What is the midpoint between points P (0, 5) and Q(5, 0)?

5/2, 5/2

Solve cot(arcsin(2/5)

5/√21

Thor divided a number by 4 and then multiplied it by 16, and lastly subtracted 22 from the new total. His total was 178. What was Thor's initial number?

50

Multiply: P(x)=5x²+x Q(x)=x+3

5x³+16x²+3x

Find the radian measure of 225⁰.

5π/4

What is the amplitude of y=-6 cosx?

6

Add the following polynomials: Q(x)=6x⁵+4x³−2x=0 P(x)=-2x²+x−6=0

6x⁵+4x³−2x²−2x−6=0

That sin a=3/5 where 90 ⁰ <a < 180⁰, find cos 2a

7/25

How many solutions does the equation cos(3x)sin(2x) =0 have on the interval [0,2π] radians?

9

For the Pythagorean Theorem to be valid, what must one of the triangle´s angles be equal to?

90°

What is the arcsin(1) in degrees?

90⁰

Divide P(x) by Q(x), where P(x)=9x³−2x²+8x−16 Q(x)=x−1

9x²+7x+15 with a remainder of 1

How is the angle of radian measure denoted?

Alpha

For a periodic function, y=Af(x), what is the absolute value of |A| known as?

Amplitude

What values satisfy ax² + bx + c > 0, where ∆ = b² -4ac = 0 and a > 0?

Any number where x ≠ b/2a

What is the inverse of the function y=sin x over the interval -90≤x≤90 called?

Arcsine

How are the sets of all complex numbers denoted?

C

which mathematical law states for all real numbers X×Y=Y×X

Communication Law of Multiplication

In the Division Algorithm, what two cases can the remainder R(x) be?

Either a zero polynomial or of a degree that is less than the degree of K(x)

True or False? You can solve for a radian measure if and only if the angle is 90⁰.

False

True or False? You can solve for a radian measure if and only if the length of the arc is formed from an obtuse angle.

False

State the Remainder Theorem

If R is the remainder when the polynomial P(x) is divided by x−a, then R=P(a)

For a circle with radius r centered at vertex a, let 1 be the length the arc intercepted by a. When is 1 considered to be negative?

If a is a negative angle

When can the period of a function change?

If the function is of the form y=f(ax) where a ≠0

What is the converse of the Pythagorean Theorem?

If the square f one side of a triangle is equal to the sum of the square of the other two sides, then the triangle is said to be a right angle.

State the Pythagorean Theorem.

In a right triangle, the square of the hypotenuse is the sum of the squares of the other two sides of the triangle.

How many possible solutions does the equation below have? 2(x−4)+3=2x−5

Infinite

Is √6 a rational or irrational number?

Irrational

A polynomial K(x) divides a polynomial P(x) if there exists a polynomial Q(x) such that P(x)=

K(x)×Q(x)

State the Division Algorithm

Let P(x) and K(x) be twp polynomials, where K(x) is not the zero polynomial. Then there exists two unique polynomials Q(x) and R(x) such that P(x) =[K(x)×Q(x)+R(x)]; the polynomial Q(x) is the quotient, and the remainder R(x) is eitehr the zero polynomial or of a degree that is less than the degree of K(x)

What is the midpoint between points formula?

M= (x₁+x₂)/2, y₁+y₂/2)

Sarah left at 8:00 am and arrive at 10:00 am. Mark left at 9:00 am and arrived at 10:30 am to the same location. Whose route took the least amount of time?

Mark

Are the following three equation equivalent? 3x²−6x+9=0; 9x²−18x=-27 and 2x=-x²−3

No

Are the following three equations equivalent? 5x+6−2x−4=0, 3x+3=6 and 3x=−2

No

In a polynomial equation, if the coefficient is zero, is there a need to write the corresponding term?

No

Is the Pythagorean Theorem valid for acute triangles?

No

Is the Pythagorean Theorem valid for obtuse triangle?

No

Is the function y=cosx considered invertible?

No

Will the remainder of the polynomial K(x)=[P(x)÷Q(x)] always be zero?

No

what is the solution of the inequality ax² + bx + c< 0, where ∆ b² _4ac < 0 and a > 0?

No solution

The polynomial x−a is a factor of P(x), if and only if

P(a)=0

What type of equation is ax²+bx+c=0, where a≠0?

Quadratic Equation

What type of number is a ratio, or division of two integers?

Rational Numbers

The Factor Theorem follow from which thereom?

Remainder Theorem

solve: cot (210°)

Root(3)

What term is used to denote the period of a function?

T

Which theorem states: Let P(x)=(A sub n)×(x^n)+a(sub n−1)×(x^n)+....+a₁x+a₀ be the polynomial with integer coefficients. If a reduced fraction (m÷k) is solution to equation P(x)=0 then m divides a₀ and k divides a(sub n)

The Rational Root Theorem

Describe the area of triangle?

The area of any triangle ABC is half the product of a side of the triangle and the altitude of that side>

What does it mean if we say the radian measure of an angle is 1.5?

The length of the arc intercepted by a in a circle of radius 1, is 1.5.

What m ust be proven to guarantee tan a is a function?

The value of tan a is independent of the choice P(x, y) on the second side of the angle.

What is the one advantage of equation y=mx+b over the equation ax + by + c = 0?

The values m and b tell us how the line is position in the coordinate system.

What does the periodic term mean in relation to the graphs of trigonometric functions?

There exists a number T for which f(x+T) =f(x) for any x in the domain range.

True or False? If you are given the radian measure of an angle you can determine the angle measurement in degrees.

True

When are 2 equations equivalent?

Twp equations are equivalent if they have the same solution set, that is to say if each solution of one equation is also the solution to the second solution an vice versa.

List 3 properties of 0 and 1, where x equals any real number?

X+(-X)=0 (1/X)=X⁻¹ ; X×X⁻¹=1

If X and Y are real numbers, and their difference X-Y, then what is the real number?

X+(-Y)

Are the following two equations equivalent? 4x-16=4 and 4x-20=0

Yes

Are the following two equations equivalent? 7x−16−3x=0 and x−4=0

Yes

Does ax² + bx + c, when a > 0 have a minimum?

Yes

Is arctan(x) the same as tan−¹ x?

Yes

Is the Pythagorean Theorem valid for right triangles?

Yes

Is y=cosx with the domain 0⁰≤y≤180⁰ invertible?

Yes

For any real number, if X<Y and Z<0, i XZ or ZY greater?

ZX>ZY

What is the expression of a complex number where a and b are real numbers?

a+bi

what is the conjugate of a-bi?

a+bi

What must a, b, and c satisfy for a line in the plain to have the equation ax + by + c = 0?

a, b , and c are real numbers and a and b are both not zero.

At what degree(s) is cot a not defined?

a=0 , a= 180° an a=360°

At which degree (s) is the tan a not defined?

a=90° and a=270°

At what degrees is cos a defined?

all values of a

Fill in the blank: The function, f(x) = 4x + 8, admits ___ value of x.

any

If the Law of Cosine holds, for triangle ABC with sides of a,b,c and angles β and y then c²=

a² + b² -2ab cos y

What is the formula for the Pythagorean Theorem?

a² + b² = c² , where side a and b of a right triangle and c is the hypotenuse.

If the Law of Cosine holds, for triangle ABC with sides a,b,c and angles β and y then b² =

a² +c² -2ac cos β

Wht is the a₂ and a₅ coefficients of the following polynomials: 6x⁵−3x⁴+x³−x+1=0

a₅=6 and a₂=0

If the Law of Cosine holds, for triangle ABC with sides a,b,c and angles β and y, then a²=

b² + c² -2bc cos a

What is the set of all points in the plane that are equal distance from a given point in the plane?

circle

Why is trigonometry used?

compute the length of segments and measure of angles

how is the cosine of an angle with 0°<a<90° denoted and defined?

cos a;cos a = sin (90° - a)

Solve: cos (α - β)

cos α cos β + sin α sin β

Solve: cos (α + β)

cos α cos β - sin α sin β

solve: cos(-a)

cosa

What two properties must hold for y= arc cos x?

cosy=x and 0⁰ <y<180⁰

solve: cos(2a)

cos²a -sin²a

solve: cos a/ sin a

cot a

How is the cotangent of an angle with 0°<a<90° denoted and defined?

cot a;cot a = tan (90° - a)

simplify the expression cosxcxscx

cotx

What is the distance between two points formula?

d= √(x₁-x₂)² + (y₁-y₂)²

What is the highest power of a polynomial called?

degree

the expression b²−4ac tells us when a quadratic equation has a real solution, and is it does, whether it has one or two real solutions. This expression is called the

discriminant

What is the set of all possible inputs of a function called?

domain

Is m considered an even or odd number, for some integer K, where m is n integer and m=2k

even

How many zeros are there for a parabola whose discriminant is equal to zero?

exactly one

How many zeros e there for a parabola whose discriminant is greater than zero?

exactly two

What term is used to denote a complex number?

i

For a circle with radius r centered at vertex a, let 1 be the length the arc intercepted by a. When is 1 considered to be positive?

if a is a positive angle

Is the ratio 1/r independent or dependent of the radius of r?

independent

What may be admitted into the domain of a function?

input

y = ax + b is what type of function?

linear

What type of equation is ax+b=0

linear equation

What type of equation is ax²+bx+c=0, where a=0?

linear equation

What is the formula for a slope of a line?

m= y₁-y₂ / x₁-x₂

When the leading coefficient of a quadratic equation is positive, the function will have a ______.

minimum

Expressions of the form x^(k), where k is non-negative integer are called

monomials

Distance lines l₁ and l₂ with equations y=m₁x+b₁ and y=m₂x+b₂ respectively l₁ and l₂ are parallel if and only if

m₁=m₂

hat condition must be satisfied for l₁ and l₂ to be two lines with non-zero slopes whose equations are y=m₁x+b₁ and y=m₂x+b₂ respectively, l₁ and l₂ are perpendicular to each other?

m₂ = 1/m₁

Do all functions admit all real values?

no

Does the experssion (√-1) represent a real number?

no

Is there is a real number for x=(±√p), where p is negative?

no

5x−6=2(x−4)+3x

no solution

7x−10=3x+6+4(x−2)

no solution

Fill in the blank. The outputs of absolute value of a function y= |x| are always ____ .

non-negative

When a discriminant of the quadratic equation equals b²−4ac<0, how many real solutions does it have?

none

Solve: |x| < 6

only numbers that satisfy -6 < x < 6

Solve: |x| > 21

only numbers that x satisfy x ≥ 21 or x ≤ 21

What may be admitted into the range of a function?

output

Fill in the blank: The diagonals of a _____ bisects each other.

parallelogram

What are the graphs of trigonometric functions considered?

periodic

What is the most basic geometric object?

point

What type of equation is 3x³−x²+4x−16=0

polynomial

What type of number cannot be factored into the product of other small integers

prime

y = ax^2 + bx + c where a ≠ 0 is what type of function?

quadratic

What is the unit measure of an angle, when measured by its length?

radians

What are the two unit measurements of angles?

radians and degrees

What is the set of all possible outputs of a function called?

range

A simplified form of an expression tan (x) csc (x) is

sec (x)

solve: 2 sin a cos a

sin (2a)

For 180° < a< 270°, where y<0 and x<0 implies what relationship for sin a and cos a?

sin a < 0 and cos a<0

For 0° < a <90°, where y > 0 and x > 0 then, is sin a > 0 or sin a <0

sin a>0

What are examples of the simplest trigonometric equations.

sin x = a , cos x = a and cot x =a

State the sum to product identity for sin(x-y)

sin x cos y- cos x sin y

Solve: sin (α + β)

sin α cos β + cos α sin β

Solve: sin (α - β)

sin α cos β - cos α sin β

What is the equation y=mx+b called where m equals the slope of line and b= y-intercept?

slope intercept form of the line.

How is the tangent of a denoted?

tan a

solve: sin a/ cos a

tan a

For 270° < a < 360, where y<0 and x>0 implies what realationship for tana and cota?

tan a < 0 and cot a a< 0

what is the slope going through the origin O and point P?

tangent of a

What is the absolute value of a complex number defined as?

the distance from z to the origin

What is important to remember when multiplying or dividing inequalities by a negative number?

the inequality sign changes

What is the goal of coordinate geometry?

to describe geometric objects algebraically through equations

What does triangle measurement mean?

trigonometry

True or False? For 90° <a < 180°, where y >0 and x<0, the sin a >0 and cos a<0.

true

When are two polynomials equal?

when they have the same terms

For all x from -90⁰<x<90⁰ solve arcsin (sinx)

x

If x is located to the right of 0, then the absolute value of x is

x

Divide P(x) by Q(x) where P(x)=x²+12x+36 Q(x)=x+6

x+6

−3x+1=x+6 (solve for x)

x= −1¼

For 6x²+11x−35=0. (solve for x)

x=(7/2, -5/3)

-2x−4=-2 (solve for x)

x=-1

Solve: 4x²−16=0

x=-2, x=2

Solve for x: 15x+60=0

x=-4

(x+4)²=16

x=0, x=-8

Solve for x: 2x=4x-2

x=1

Solve: x²−4x=-4

x=2

Solve: x²−6x+9

x=3

Solve: x²−x−6=0

x=3, x=-2

what is the formula for the quadratic equation?

x=[-b±√(b²−4ac)]÷[2a]

x²=144

x=±12

For x²−48+0. (solve for x).

x=±4√3

x²=36

x=±6

What is the real soution(s) fro the quadratic equation with the discriminant b²−4ac=0?

x=−b/2a

Divide P(x) by Q(x), where P(x)=x³−3x²+3x−1 Q(x)=x−1

x²−2x+1

Multiply: P(x)=x³−x²+x Q(x)=x⁷+x⁵+x³

x¹⁰−x⁹+2x⁸−x⁷+2x⁶−x⁵+x⁴

For all y from -1≤y≤1, solve cos (arccosy)

y

When a periodic function is of the form y= Af(x) its graph expands or shrinks along which axis?

y-axis

What does b, represent in the equation y=mx+b?

y-intercept

What is the ratio of sin a?

y/r with P(x, y) as the point the second side of the angle and r the distance between P and the origin.

What value(s) of y and for x= to any real number satisfies the equation x×y=0

y=0

Is it possible to use trigonometric functions to solve for negative angles?

yes

what is the equation of a non-vertical ie that goes through two distinct points P (x₁,y₁) and Q (x₂,y₂)?

y−y₁ = y₂- y₁ / x₂- x₁ (x- x₁)

What is a polynomial called when all of its coefficients are zero?

zero polynomial

How is the absolute value of a number x denoted?

|x|

What are all possible solutions to an absolute value?

|x| ={x if x > 0 |x| ={0 if x =0 |x| ={x if x < 0

Using the quadratic function formula solve: x^2 - 1

±1

What values does the identity cos (α -β) hold to be true?

α andβ of any measure

What is the radian measure of a, where the degree of the angle of a is 180⁰?

π

What is the radian measure of a, where the degree measure of an angle a is 1⁰?

π/180

What is the radian measure of a, where the degree of the angle a is 1⁰?

π/180

What is the radian measure of a, where the degree measure of the angle a is n⁰?

π/180 n

Find the radian measure of 90⁰.

π/2

What is the arccot(0) in radians?

π/2, 3π/2

What is the arctan 1 in radians?

π/4, 5π/4

Find the radian measure of 30⁰.

π/6

Solve arccos 1/2-arccos √3/2, in radians

π/6

What is the arcsin(1/2) in radians?

π/6, 5π/6

Solve sin(arcsin π/7)

π/7

What symbol is used to denote the discriminant of the quadratic equation?

∆

State the cos a/2 half angle identity.

√1+cos a/2

Solve sin 45⁰

√2/2

What is the distance between points P (-1, 3) an Q (10, 15)?

√265

Solve: sin (120°)

√3/2

solve: sin 60°

√3/2

solve: sin90°*cos30°

√3/2

Evaluate: cos 60° sec30°

√3/3