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Solve trigonometric equations

Pick the story that can be modeled by the equation

x - 6 = 2

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(A)Last week, Anthony borrowed

x

books from the library. Today, he returned

6

of the books. He wasn't finished with the remaining

2

books, so he renewed them.

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(B)Last week, Anthony borrowed

6

books from the library. Today, he returned

x

of the books. He wasn't finished with the remaining

2

books, so he renewed them.

Pick the story that can be modeled by the equation

2 + x = 8

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(A)During yesterday's lacrosse game, Maya's team scored a total of

8

goals. They scored

2

goals in the first half and

x

goals in the second half.

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(B)During yesterday's lacrosse game, Maya's team scored a total of

x

goals. They scored

2

goals in the first half and

8

goals in the second half.

Katy's class visited the bookfair this afternoon. She bought a set of

m

books from her favorite mystery series and a set of

4

books from her favorite science fiction series. Katy bought

9

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Which diagram models the story?

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Which equation models the story?

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m + 4 = 9

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m - 4 = 9

Solve the following system of equations using an inverse matrix. You must also indicate the inverse matrix,

A^{-1}

, that was used to solve the system. You may optionally write the inverse matrix with a scalar coefficient.

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\begin{cases} -2x-5y=-6 \ -x+y=8 \end{cases}

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A^{-1}=\square[\square]\quad x=\square\quad y=\square

Find all solutions with

- \frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}

. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.

\csc(\theta)=1

Find all solutions with

0\leq\theta\leq\pi

. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.

– 7\sec(\theta)–7=0

Find all solutions with

- \frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}

. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.

9\csc(\theta)-18=0

Find all solutions with

- \frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}

. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.

14\csc(\theta)+14=0

Find all solutions with

- \frac{\pi}{2} < \theta < \frac{\pi}{2}

. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.

- 7\tan(\theta) + 7 = 0

Find all solutions with

- 90°<\theta<90°

. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.

- 8\tan(\theta)–8=0

Find all solutions with

- 90°<\theta<90°

. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.

2\tan(\theta)=0

Find all solutions with

-90^\circ \leq \theta \leq 90^\circ

. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.

- 2\sin(\theta)+ \frac{9}{2} = - 13\sin(\theta)+10

Find all solutions with

- 90°\leq\theta\leq90°

. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.

- \sin(\theta)= - 3\sin(\theta)

Find all solutions with

0\leq\theta\leq\pi

. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.

-14\cos(\theta)+7=0

Find all solutions with

0\leq\theta\leq\pi

. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.

-14\cos(\theta)+7=0

A

, of the region between the curve

y=\frac{2x}{1+x^{2}}

and the interval

-2 \leq x \leq 2

x

-axis. The area is

A=\square

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(Type an exact answer.)

m

b

are constants, which equation represents a linear function where

x

is the input and

y

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y = \frac{m}{x} + b

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y = mx + b

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y = mx^2 + b

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Which statement about the function is true for any values of

m

b

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(A)The graph has a constant rate of change.

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(B)The graph has a positive

y

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(C)The graph has a positive slope.

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(D)The graph is not a straight line.

Which formulas represent linear relationships? Select all that apply.

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Multi-select Choices:

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(A)the area of a semicircle,

A = \frac{\pi r^2}{2}

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(B)the volume of a square pyramid with a height of

1

V = \frac{1}{3}s^2

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(C)the perimeter of a regular pentagon,

P = 5s

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(D)the volume of a hemisphere,

V = \frac{2}{3} \pi r^3

Which formulas represent linear relationships? Select all that apply.

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Multi-select Choices:

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(A)the area of a triangle whose height and base are the same,

A = \frac{1}{2}h^2

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(B)the perimeter of a regular octagon,

P = 8s

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(C)the surface area of a cube,

SA = 6a^2

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(D)the perimeter of a semicircle,

P = \pi r + 2r

Ann solved the equation

9 + 2n - 3 = \frac{1}{2}(4n + 6)

. Here are her last two steps:

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6 + 2n = 2n + 3

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6 = 3

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Which statement is true about the equation?

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(A)The solution is

n = \frac{6}{3}

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(B)There is no solution because

6 = 3

is a false equation.

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(C)There are infinitely many solutions because

6 = 3

is a true equation.

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(D)The solution is

(6, 3)

Haley solved the equation

25 + 3n - 12 = 13 + 7n - 4n

. Here are her last two steps:

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13 + 3n = 13 + 3n

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13 = 13

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Which statement is true about the equation?

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(A)The solution is

n = 13

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(B)There is no solution because

13 = 13

is a true equation.

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(C)There are infinitely many solutions because

13 = 13

is a true equation.

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(D)The solution is

(13, 13)

Jill solved the equation

8 - p + 7 = 7p - 5 - 3p

. Here are her last two steps:

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20 = 5p

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4 = p

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Which statement is true about the equation?

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(A)The solution is

p = 4

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(B)There is no solution.

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(C)There are infinitely many solutions.

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(D)The solution is

(20, 4)

Which of these equations has no solutions?

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\frac{1}{3}(9x + 3) = 4x + 5 + x

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-3(x + 5) = -x - 2x - 15

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-2x + 5 + 8x = 3(2x - 1)

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Which statement explains a way you can tell the equation has no solutions?

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(A) It is equivalent to an equation that has the same variable terms but different constant terms on either side of the equal sign.

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(B) It is equivalent to an equation that has the same variable terms and the same constant terms on either side of the equal sign.

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(C) It is equivalent to an equation that has different variable terms on either side of the equation.

Which of these equations has infinitely many solutions?

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2(x + 3) = 4x + 6

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3x - 5 + 2x = 5(x - \frac{1}{5})

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-2x + 3 + 4x = \frac{1}{2}(4x + 6)

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Which statement explains a way you can tell the equation has infinitely many solutions?

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(A) It is equivalent to an equation that has the same variable terms but different constant terms on each side of the equal sign.

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(B) It is equivalent to an equation that has the same variable terms and the same constant terms on each side of the equal sign.

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(C) It is equivalent to an equation that has different variable terms on each side of the equation.

Which of these equations has no solutions?

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\frac{1}{4}(12x + 16) = 5x + 4 - 2x

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3x + 1 - 5x = 4x - 11

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4(2x - 1) = 8x - 1

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Which statement explains a way you can tell the equation has no solutions?

\newline

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(A) It is equivalent to an equation that has the same variable terms but different constant terms on either side of the equal sign.

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(B) It is equivalent to an equation that has the same variable terms and the same constant terms on either side of the equal sign.

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(C) It is equivalent to an equation that has different variable terms on either side of the equation.

Devin solved the equation

12 + 5x - 7 = 13x - 4 - 8x

. Here are his last two steps:

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5 + 5x = 5x - 4

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5 = -4

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Which statement is true about the equation?

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(A)The solution is

x = -4

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(B)There is no solution because

5 = -4

is a false equation.

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(C)There are infinitely many solutions because

5 = -4

is a false equation.

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(D)The solution is

(5, -4)

Solve this system of equations by graphing. Graph the equations, then find the solution.

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y = -x - 5

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y = -\frac{1}{2}x - 3

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Click to select points on the graph.

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The solution is (

\_, \_

How many solutions does the system of equations below have?

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y = \frac{8}{7}x + 10

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y = \frac{8}{7}x + 10

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(B)one solution

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(C)infinitely many solutions

How many solutions does the system of equations below have?

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y = 6x + 10

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y = 6x + 10

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(A) no solution

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(B) one solution

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(C) infinitely many solutions

How many solutions does the system of equations below have?

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y = -x - 2

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y = -x - 2

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(B)one solution

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(C)infinitely many solutions

How many solutions does the system of equations below have?

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y = x + 6

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y = x + \frac{9}{4}

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(B)one solution

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(C)infinitely many solutions

How many solutions does the system of equations below have?

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y = 9x - \frac{2}{9}

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y = 9x - \frac{2}{9}

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(B)one solution

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(C)infinitely many solutions

How many solutions does the system of equations below have?

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y = -5x + 4

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y = -5x + \frac{5}{2}

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(B)one solution

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(C)infinitely many solutions

Abby and Robert are each trying to solve the equation

x^{2}-10x+26=0

. They know that the solutions to

x^{2}=-1

i

-i

, but they are not sure how to use this information to solve for

x

in their equation. Solve the equation and explain to them where the

i

f(x)=(2 x+5)(x+3)(3 x-10)(2 x-8)

x=-3, x=-\frac{5}{2}, x=\frac{10}{3}

x=4

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What is the sign of

f

on the interval

-\frac{5}{2}<x<4

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1

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f

is always positive on the interval.

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f

is always negative on the interval.

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f

is sometimes positive and sometimes negative on the interval.

f(x)=(2 x-3)(x+6)(5 x+6)

x=-6, x=-\frac{6}{5}

x=\frac{3}{2}

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What is the sign of

f

on the interval

-6<x<-\frac{6}{5}

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1

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f

is always positive on the interval.

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f

is always negative on the interval.

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f

is sometimes positive and sometimes negative on the interval.

f(x)=(x-2)(x+4)(x+5)(x-1)(x-9)

x=-5, x=-4, x=1, x=2

x=9

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What is the sign of

f

on the interval

-4<x<1

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1

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f

is always positive on the interval.

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f

is always negative on the interval.

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f

is sometimes positive and sometimes negative on the interval.

z=20i+26

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What are the real and imaginary parts of

z

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1

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\text{Re}(z)=20

\text{Im}(z)=26

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\text{Re}(z)=26

\text{Im}(z)=20i

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\text{Re}(z)=20i

\text{Im}(z)=26

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\text{Re}(z)=26

\text{Im}(z)=20

w

is rational. Which statement about

w - 6

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w - 6

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w - 6

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w - 6

can be rational or irrational, depending on the value of

w

If the simultaneous linear equations in

x

g

y = kx + m

y= (2k - 1)x +4

have at least one solution, where

k

m

are constants, find the possible values of

k

m

Find the area of the region bounded by the curves

y=\sqrt{x}

y=x^{2}

, and the lines

x=0

x=1

\mathrm{Q}

-14

\mathrm{R}

\mathrm{S}

9

units away from Point

\mathrm{Q}

\mathrm{R}

\mathrm{S}

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\mathrm{R}=\square \quad \mathrm{S}=\square

Garret makes a ramp for his skateboard in the shape of a right triangle with a hypotenuse of

2\,\text{ft}

1\,\text{ft}

. He wants to use a trigonometric ratio to describe the relationship these two sides. Select all of the expressions that he could use.

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\sin 30^{\circ}

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\cos 45^{\circ}

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\tan 30^{\circ}

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\sin 45^{\circ}

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\cos 60^{\circ}

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\tan 45^{\circ}

Garret makes a ramp for his skateboard in the shape of a right triangle with a hypotenuse of

2 \mathrm{ft}

1 \mathrm{f}

. He wants to use a trigonometric ratio to describe the relationship these two sides. Select all of the expressions that he could use.

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\sin 30^{\circ}

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\cos 45^{\circ}

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\tan 30^{\circ}

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\sin 45^{\circ}

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\cos 60^{\circ}

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\tan 45^{\circ}

How many solutions does the system of equations below have?

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y = 2x - 1

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y = 2x - 1

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(B)one solution

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(C)infinitely many solutions

How many solutions does the system of equations below have?

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y = 2x + \frac{5}{9}

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y = 2x + \frac{5}{9}

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(B)one solution

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(C)infinitely many solutions

How many solutions does the system of equations below have?

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y = \frac{2}{5}x + 6

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y = \frac{2}{5}x + 6

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(B)one solution

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(C)infinitely many solutions

How many solutions does the system of equations below have?

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y = -x + 5

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y = -x + 5

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(B)one solution

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(C)infinitely many solutions

How many solutions does the system of equations below have?

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y = \frac{2}{3}x - 3

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y = \frac{2}{3}x - \frac{5}{6}

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(B)one solution

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(C)infinitely many solutions

How many solutions does the system of equations below have?

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y = 5x + 9

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y = 5x + 9

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(B)one solution

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(C)infinitely many solutions