x-7=1

Correct: y = 0.63x + 1.90

(4,-7), (-2,-10)

Solution

Slope Formula:

slope(m) = (y2 – y1) / (x2 – x1)

Substitute and Solve

m = (-10 +7) / (-2 – 4)

m = (-3) / (-6)

m = 1/2 or 0.5

Slope intercept form:

y = mx + b

Pick one of the two points (4,-7) or (-2,-10), substitute and solve

-7 = 1/2(4) + b

-7 = 2 + b

-7 – 2 = b

-9 = b

Final Answer

y = 1/2x – 9

y = -4x + 5

Formula: y = ky(solving for y) or x = ky(solving for x)

Substitute

78 = k(130)

Divide by 130 on both sides

k = 0.6

x = 0.6(190)

x = 114

a. Write a function expressing the cost, C(x), of the project in terms of the number of stuffed animals decorated, x.

b. Determine the cost of decorating 25 stuffed animals.

c. How many stuffed animals can be decorated with a budget of $227.75?

b. $175.25

c. 35

f(x) = 6 – x, g(x) = x² – 2

Solution

(6-x)² – 2

36 – 6x – 6x + x² – 2

x² – 6x – 6x + 36 – 2

x² – 12x + 34

(f o g)(x) = -x² + 8

Solution

-x(x² – 2) + 6

-(x² – 2) + 6

-x² + 2 + 6

-x² + 8

((-u)³(-u⁹)⁸)/(u⁵)³

Solution

(-u³ × u⁷²)/u¹⁵

-u⁷⁵/u¹⁵ = -u⁶⁰

A(x) = (3.75x + 200)/x, where x is the number of T-shirts made. Find the average cost per T-shirt when the company makes 10 shirts.

(13 + 3)+ 8 = 13 + (3 + 8)

{7x – 3y = -43

{5x + 6y = -47

{x = 8t – 4

{y = 6t² – 2

Solution

x = 8t – 4

Add 4 to both sides

x + 4 = 8t

Divide by 8 on both sides

t = (x + 4)/8

Substitute

y = 6((x+4)/8)² – 2

F + C ≤ 30,775

C ≥ 7F

F ≥ 0; C ≥ 0

{3x + 4y = 32

{3x + y = 17

{x + y = -1

{y = 3x + 19

{3x + 3y = 2

{3x – y = 8

{2x – 5y = 1

{5x -2y = -4

{-16x – 6y = 170

{8x + 3y = -85

Correct: Infinitely Many Solutions

{x + y = -4

{y = 3x – 8

{2x + 3y < 1 {x -3y > 2

(3, 1), (-3, -2), (5, -5), (-2, 4), (-4, 5)

y < 3x - 2 y > -2x + 5

Use mathway; Graph the intersection (and)

Constraints:

{x + y ≥ 1

{4x – 2y ≤ 4

{3y ≤ 3x + 3

Solution

Substitute

3(5) – 4 ≤ 15

11 ≤ 15

True

Solution

Substitute

-2(1) – 5(7) ≥ -32

-2 – 35 ≥ -32

-37 ≥ -32

False, not a solution of the inequality

{4x + 2y = -12

{3x – 2y = 5

Solution

x(t)=350t

y(t)=3.21-14t

Solve either equation for “t”

x=350t

t=x/350

Substitute

y=3.21-14(x/350)

Substitute for “x”, x=11

y=3.21-14(11/350)

y=2.77

Constraints:

{x + y ≥ 2

{6x – 6y ≤ 12

{8y ≤ 4x + 16

Correct: The maximum value of C is 106 at (8,6) and the minimum value is 10 at (2,0).

{S + B ≤ 44,000

{S ≥ 0

{B ≥ 0

{4y = 4

{x + 2y + z = 10

{y – 2z = -5

Correct:

x=5

y=1

z=3

{x = 8t² -3

[y = 6t – 7

[1 17 14][11 3]

[19 7]

[9 5]

matrix equation. If not possible, classify the system.

{4x + y = -24

{2x + 2y = -18

You can just solve by using Elimination or Substitution

det [10 -7]

[-5 -6 ]

{x – y = -9

{5x = 0

[5 0| 0]

Solution

x=1, y=-1

5x=5

[10 16 8] [12]

[ 2 ] =

[216]

[258]

[3 -6 -7] [-2 -1 7]

[-3 -2 4] + [ 3 5 -7]

[-4 -1 7] [4 -9 -3]

[ 0 3 -3 ]

[0 -10 4]

{8x + 2y + 6z = -3

{3x – y + 10z = 3

{-9x -2y -4z = -7

[3 -1 10| 3 ]

[-9 -2 -4 |-7 ]

[ 6 2 ]

[16 32 55] [ 8 11]

[18 16]

det[x+3 2]

[3 x-2]

=0

Correct: x = 0

Mathway: Find the Determinant

{-3x -7y -2z = -25

{-6x +21y -6z = 57

{-3x +14y -4z = 40

{-6x + 9y = -5

{-8x – 4y = 0

[1 0| 5/24 ]

[0 1|-5/12]

Correct:

[1 0| -5/24]

[0 1| 5/12

[-8 -8] + [-2 3]

[-6 4] + [3 -1]

(2×2 matrice + 2×2 matrice)

[-3 3]

[-12 -44]

{3x + 2y + z = -2

{3x -2y -z = -22

{3x + 2y -z = -10

[3 -2 -1] [y] = [-22]

[3 2 -1] [z] [-10]

;(-4, 3, 4)

[2 -13 -6]

[0 5 0]

[0 -24 0]

{x + 2y + z = -17

{3x + 7y + 2z = -56

{x – y + 2z = 0

Victor [24 -17]

Wanda [28 -12]

[2 -3 -3 2]

[6 2 -7 3]

2x² + 5x -3 = 0

Use Quadratic equation:

(-b ± √((b² – 4ac))/2a

(1, 4.1), (5, 2.3), (11, 4.9)

Mathway:

Find the Equation of the Parabola

y = -6x² – 24x – 31

y = 6(x + 2)² – 7

x = -2; (2,-7)

Correct:

y = -6(x + 2)² – 7

x = -2; (-2, -7)

Mathway:

Find the Vertex Form

Find the Axis of Symmetry

Find the Vertex (Coordinates)

6x² – 24x – 30 = 0

Correct: -4 ± 2√6; 0.9 or -8.9

x² – 2x ≥ 24

Correct:

y = (x+7)² – 45

x = -7; (-7, -45)

x² – 4x + 3 < 0

distance, x | 8 | 18 | 33 | 43 |

height, y | 65 | 76 | 79 | 71 |

Choose the quadratic equation that best fits the baseball’s trajectory from Gupta to the open field below.

f(x) = (5x – 4) (5x + 3)

a = 25, b =5, c = -12

Graphing Calculator

STAT→Edit

Put x values into L1 and y values into L2.

2ND→MODE→STAT→CALC

→5:QuadReg

In(8x – 6) = 3

x ≈ 3.26

Exponential form

e³= 8x – 6

Correct: x=-1/3

Solution

log(x) – log(y) = log(x/y)

log₇ 8 – log₇ (x+3) = log₇ (8/(x+3))

log₇ (8/(x+3)) = log₇ 3

(Mathway can solve once the left terms are combined)

Correct: M = 8.4

Solution

M = 2/3 log₁₀ (10^24.4/10^11.8)

Correct: “b” has to be greater than 0 and less than 1. (0 < b < 1)

Solution

2.37(1+0.031)⁹

Correct: 0.9995

Solution

100 – 0.05 = 99.95

99.95/100 = 0.9995

6^-4 = 1/1296

log₈ 1/2

f(x) = x⁴ – 9x³ + 10x² + 90x – 200

x³ – 7x² + 14x – 8 = 0

f(x) = x² – 10x + 34

Correct: 5 ± 3i

x³ – x² – 14x + 24 = 0

f(x) = x³ + 4x² + 6x + 9

(5x² – 2x + 5)(x² – 8x – 2)

P is of degree 3; P(0) = 24; zeros: 3, -4, 2

f(x) = x³ – 4x² + 7x – 12

P(x) = 3x⁴ – 10x³ – 41x² + 68x + 60

(x – 6)(x + 5)(x – 4)

Mathway (Algebra)

Factor

f(x) = 3x⁴ – 2x³ + 2x + 2

f(x) = x⁴ – x³ – 16x² + 10x + 60

f(x) = (x – 1)/((x – 7)(x + 3))

rational; x≠1, 7 or -3; asymptotes at y=0, x=1, x=7 and x=-3

Correct:

rational; x≠-3,7

asymptotes at y=0, x=-3, and x=7

( 3/(x² + 7x + 12) + 1/(x² + 11x + 28) ) ÷ ( 2/(x² + 10x + 21) + 1/x² + 13x + 36) )

( (x – 2)/(2x²) ) – ( (2x + 1)/(9x) ) + ( (7x)/(12) )

³√(3x-6) + 6 ≥ 7

( (x² + 14x + 49)/8x ) ÷ ( (x+7)/4x )

2 ÷ √5

(√11) ÷ (9 + √5)

(-3x + 8)/(x² – 49) – (-2x + 1)/(x² – 49)

y=-99.4 when x=-11 and z=4. Find y when x=2 and z=-8.

³√(-27x) – 4 ³√(x⁴) + 9 ³√(x) + 9x ³√(x)

4/(x-3) – 2/(x+4) = 0

Solution

f(x) – √(9-x) = 0

Add √(9-x) to both sides

f(x) = √(9-x)

Mathway: Find the Domain

y varies jointly as x and z. y=98/3 when x=2 and z=7. Find y when x=8 and z=3

A group of college students are volunteering for Habitat for Humanity during their spring break. They are putting the finishing touches on a house they built. Working alone, Dale Horton can paint a certain room in 3 hours. Kathy Garcia can paint the same room in 9 hours. How long will it take them working together to paint the room?

Correct: W (wattage) = 24

(2x-2)/(x²-36) ≥ 1/(x+6)

√(x + 14) ≤ x – 16

vertices: (2, 0) and (-2, 0)

asymptote: y=1/2x

Put in each answer using

Mathway: Find the Vertex Form, till the verices and asymptote matches up

Diameter CD, C(-1, -4), D(5, 4)

radius: 5

2x² + 2y² – 28x + 24y + 162 = 0

Diameter CD, with endpoints C(- 1, 5), D(5, 9)

radius: √13; 3.61

Mathway: Find the Midpoint

radius: 7

center: (1, 1)

{x² + y² = 144

{x² – 4y² = 64

x = 1/24 y²

directrix: x=-6

{x² + y² = 256

{x + y = 16

x = -1/4 y²

focus: (-1, 0)

vertex: (0, 0)

Mathway: Graph

center: (5, 4)

one focus: (-25, 4)

one vertex: (23, 4)

x²/9 – y²/25 = 1

vertices: (2, 0) and (−2, 0)

asymptote: y = 3x

vertex: (-1, 2)

radius: 4

center: (3, – 5)

Mathway: Find the Equation (put coordinate points comma r=4)

midpoint = (3, 9/2)

Mathway

vertices: (0, ±8)

foci: (0, ±√55)

((₁₀C₄ × ₁₂C₁)+(₁₀C₅ × ₁₂C₀))÷(₂₂C₅)

((210 × 12)+(252 × 1))÷(26334)

2772÷26334 = 0.105

Solution

Probability of getting chosen is 5/33, so the the probability of not getting chosen is 33 – 5 = 28

Solution

₄P₄

4 × 2 × 5 = 40

Solution

1/52 × 4/51 = 0.0015082

Solution

₁₂C₇ = 792

₁₄C₁₀ = 1001

₁₃C₅ = 1287

(792 × 1001)/1287 = 616

Solution

10/27 × 10/26 = 0.1424501

Solution

(n-1)!

(5-1)!

4! = 4 × 3 × 2 × 1 = 24

Solution

(n-1)!

(6-1)!

5! = 5 × 4 × 3 × 2 × 1 = 120

Solution

52 choose 3

₅₂C₃ = 22,100

Solution

nCr = n!/(r!(n-r)!)

23!/(14!(23-14)!)

23!/(14!(9)!)

23!/ (9! 14!)

H E H E G W I

Solution

Probability of selecting “E” or “G” is 3/7.

7/7 – 3/7 = 4/7

Correct: 0.106 or 47/442

Solution

(13/52 × 12/51) + (12/52 × 11/51) – (3/52 × 2/51)

Solution

t(n) = t(1) + (n – 1)d

102 = -3 + (6 -1)d

102 = -3 + 5d

105 = 5d

d = 21

Substitute

t(2) = -3 + (2 – 1)21

-3 + (1)21

-3 + 21

18

t(3) = -3 + (3 – 1)21

-3 + (2)21

-3 + 42

39

And so on

Correct: 5/16 (10/32)

Solution

(n C k)(p∧k)(1 – p)∧(n-k)

(₅C₃)(½)³(1 – ½)⁵⁻³

(₅C₃)(½)³(1 – ½)²

10 × 1/8 × 1/4 = 5/16

Mathway: Expand Using the Binomial Theorem

Solution

S = 0.2/(1 – 0.1)

S = 0.222… or 2/9

Solution

Use nCk-₁ to love for nth entry

₁₂C₆-₁ = ₁₂C₅ = 792

₁₂C₈-₁ = ₁₂C₇ = 792

Correct: 35

Solution

₇C₃ = 35

Solution

t₁=6 and t₅=1875/8

t(n) = t₁r^(n-1)

t₅ = 6r⁴

1875/8 = 6r⁴

r = 2.5

t₂ = 6(2.5)¹

15

t₃ = 6(2.5)²

37.5 or 75/2

t₄ = 6(2.5)³

9375 or 375/4

Solution

t₁ = 30 and d = 1

t(n) = t₁ + (n-1)d

t₅₀ = 30 + (49)1

t₅₀ = 79

S = n/2[2t₁ + (n – 1)d]

S = 50/2[2(30) + 49]

S = 25(60 + 49)

S = 25(109)

S = 2725

3 – 9/5 + 27/25 – 81/125 + 243/625 – …

Solution

(9/5) ÷ 3 = 0.6

r = 0.6 (common ration)

t₁ = 3

S = t₁ ÷ (1 – r)

S = 3 ÷ (1 – 0.6)

S = 3 ÷ 0.4

S = 7.5 or 15/8

t(n) = 12 – 3n

Solution

t(1) = 12 – 3(1)

9

t(2) = 12 – 3(2)

6

…

Correct: ≈ 0.072

Solution

(nCk) × (p)^n × (1 – p)^(n – k)

(₈C₃) × (0.14)³ × (1 – 0.14)⁵

= 0.072

7/4, 49/24, 343/144, 2401/864, …

Solution

explicit formula for nth term in a geometric sequence is

t(n) = t₁r^(n-1)

t₁ = 7/4 and r = 7/6

t(n) = 7/4(7/6)^(n-1)

(Evernote)

∑3(-½)^(k-1)

Solution

∑3(-½)^(k-1)

t₁ = 3(-½)⁰

3

t₂ = 3(-½)¹

-1.5

r = t₂/t₁

r = -1.5 ÷ 3

r = -½

S = t₁/(1 – r)

S = 3/(1 – (-½))

S = 3/1.5

S = 2

Solution

r ≠ 1

r = 8/3 ÷ 8

r = 1/3 & t₁ = 8

S(n) = t₁[(1 – r^n) ÷ (1 – r)]

S₇ = 8[(1 – (1/3)⁷) ÷ (1 – 1/3)]

S₇ = 11.99

Correct: 4

3

3 + 6 + 3

3 + 6 + 9 + 6 + 3

3 + 6 + . . . + ( 3n − 3) + ( 3n ) + ( 3n − 3) + . . . + 3

Correct: The sum of the rows are 3, 12, 27, . . . . The sum of the nth row is 3(n²). The sum of the 7th row is 147.

Correct: 56, 448

Solution

t₁=7 and t₄=3584

Formula

t(n) = t₁r^(n – 1)

Substitute

t(4) = 7r⁴⁻¹

3584 = 7r³

divide both sides by 7

r³ = 512

Take cube root of both sides

r = 8

t(2)=7 × 8

56

t(3)=7 × 8²

448

7, 56, 448, 3584

Median: 16

Mode: 7, 16

29, 14, 40, 12, 16, 46, 22, 23, 19, 31, 7, 9

Maximum = 46

Q₁=13; Q₂=20.5; Q₃=30

Range = 39

IQR = 17

Q₂ = mean

Solution

(Graphing calculator)

2nd→VARS→DRAW→1:ShadeNorm(→Enter→75, 90, 85, 7)

Between, mean, then standard deviation

Solution

(Graphing calculator)

2nd→VARS→DRAW→1:ShadeNorm(168, 10⁹⁹, 201, 33)→Enter = 0.84

.84 or 84%

84% of 120 is 100.8

rounded to the nearest whole number equals 101

7, 20, 7, 15, 21, 4, 22, 17, 13

https://www.easycalculation.com/statistics/standard-deviation.php

Take the Variance (Population Standard deviation) and Population Standard deviation

27, 8, 9, 48, 19, 10, 29, 42, 2, 50, 44

Maximum = 50

Q₁=9; Q₂=27; Q₃=44

Range = 48

IQR = 35

Mathway

Find the Five Number Summary, range, and Find the Interquartile Range (H-Spread)

Correct: 1.82 (Close to a “C”)

Correct: 31/100 or 31% (maybe 0.315 or 32%)

Solution

.11 + .20 + .31 + 30 + .15 + 24 = 1 (or 100%)

Age 1= .20

Age 2= .11

.20 + .11 = .31 (or 31%)

31/100 or 31% Probability

5, 6, 1, 5, 5, 2, 6, 1, 2, 2

Find the range and the mean deviation of the data.

To find mean deviation minus the mean from each data value, add everything up, then divide by the number of data values.

Example:

54, 49, 47, 48, 52

mean of data = 50

54 – 50 = | 4 | = 4

49 – 50 = |-1| = 1

47 – 50 = |-3| = 3

48 – 50 = |-2| = 2

52 – 50 = | 2 | = 2

4 + 1 + 3 + 2 + 2 = 12

12 ÷ 5 = 2.4

Mean deviation = 2.4

Solution

((₆C₁)(0.06)¹(1 – 0.06)⁵) + ((₆C₀)(0.06)⁰(1 – 0.06)⁶)

= 0.954

Solution

(₆C₃)(0.67)³(1-0.67)³

= 0.21617

Median = 17 in

Mode = 13

The median is the most useful.

Solution

(Graphing calculator)

2nd→VARS→DRAW→1:ShadeNorm(525, 589, 525, 64)→Enter = 0.3413

((₇C₁)(0.06)¹(1 – 0.06)⁶) + ((₇C₀)(0.06)⁰(1 – 0.06)⁷)

≈ 0.938

Solution

z-score

z = (x – x̅) ÷ σ

z = (83 – 77.5) ÷ 5.8

= 0.948 → 0.95

(Graphing calculator)

2nd→VARS→DRAW→1:ShadeNorm(-10⁹⁹, 83, 77.5, 5.8)→Enter = 0.83 or 83%

Solution

z-score

z = (x – x̅) ÷ σ

z = (37 – 30) ÷ 5.5

= 1.27

(Graphing calculator)

2nd→VARS→DRAW→1:ShadeNorm(-10⁹⁹, 37, 30, 5.5)→Enter = 0.8984 or 89.80%

Solution

tan x = opp./adj.

24/32 = 0.75

Solution

π = 180°

-π/6

-180°/6 = 30°

Mathway

Evaluate (Do not use Find the Exact Value)

sinθ = opp./hyp.

Solution

We have two angles

m∠A + m∠B + m∠C = 180

m∠A + 28° + 90° = 180°

m∠A = 62°

Use Law of Sines to solve for side “a” or “b’

a/sin A = c/sin C

a/sin(62°) = 18/sin(90°)

a = 18sin(62°)/sin(90°)

a = 15.89 → 15.9

b/sin B = a/sin A

b/sin(28°) = 15.9/sin(62°)

b = 15.9sin(28°)/sin(62°)

b = 8.45 → 8.5

Mathway (Trigonometry)

Find the Reference Angle

Mathway (Trigonometry)

Find the Exact Value

Solution

Use Law of Cosines for “c”

c = √(a² + b² – 2ab cos C)

c = 37

a = 12, b = 35, and c = 37

Use Law of Sines

sin A/a = sin C/c

sin A/12 = sin(90°)/37

sin A = 12sin(90°)/37

sin A ≈ 12/37

Convert from radians to degrees ∠A = 18.58° → 19°

Mathway (Trigonometry)

Find the Exact Value

Quadrant II, sinθ = 2/7

tanθ = -2√5/15

cotθ = -3√5/2

secθ = -7√5/15

cscθ = 7/2

Mathway (Trigonometry)

Find the Other Trig Values in Quadrant II

Mathway (Trigonometry)

Find the Exact Value

Mathway (Trigonometry)

Convert from Degrees to Radians

Mathway (Trigonometry)

Find the Reference Angle

Solution

120° is in Quadrant II

Coordinates:

(cos θ = x/r, sin θ = y/r)

θ = 120° and r = 13

cos(120°) = x/13

-½ = x/13

x = -13/2 or -6.5

sin(120°) = y/13

√3/2 = y/13

y = 13√3/2 or 11.26

P(-6.5, 11.26)

Correct: 3/2π

Solution

d = A cos wt

d = -6 cos 3t

w = 3

f (frequency) = w/2π

f = 3/2π

Solution

r = 4 and θ = π/30

s = rθ

s = 4(π/30)

s = 2π/15

Solution

A = displacement

d = A cos wt

d = 9 cos 2t

A = 9 or 9m

Mathway (Trigonometry)

Find the Exact Value

Solution

r = 8 and θ = π/6 (change 30° to radians)

s = rθ

s = 8(π/6)

s = (4/3)π or 4π/3

Graph all multiple choices and pick the one that fits

cos = -√3/2

tan = -√3/3

Mathway (Trigonometry)

Find the Exact Value

low tide at 6:00 a.m. and 6:00 p.m., depth 21 ft

Correct: high tide at 12:00 noon and 12:00 midnight, depth 21 ft

low tide at 6:00 a.m. and 6:00 p.m., depth 29 ft

Solution

∠A + 61° + 76° = 180°

∠A = 43°

Use Law of Sines

a/sin(43°) = 41/sin(76°)

a = 41sin(43°)/sin(76°)

a = 28.8

cos(A/2) = √3/5

sin(A/2) = 2/5

Correct:

cos(A/2) = (2√5)/5

sin(A/2) = √5/5

Solution

Pathagorean Identities

sin A = 4/5

cos²A = 1 – sin²A

cos A = 1 – (4/5)²

cos A = 3/5

Use Half-Angle Identities

sin(A/2) = √((1 – cos A)/2)

sin(A/2) = √((1 – 3/5)/2)

sin(A/2) = √5/5

cos(A/2) = √((1 + cos A)/2)

cos(A/2) = √((1 + 3/5)/2)

cos(A/2) = (2√5)/5

A = θ and the cos((1/2)A) is the same as cos(A/2)

Correct: No Solution

Solution

sin B/b = sin C/c

sin B/43 = sin(80°)/45

sin B = 43sin(80°)/45

sin B = 0.94103

sin⁻¹(sin B) = ∠B

sin⁻¹((43sin¹(80°)/45))

sin⁻¹(0.94103) = 70.2° (rounded to the nearest tenth)

∠B = 70.2°

∠A + 70.2° + 80° = 180°

∠A = 29.8°

Use graphing calculator on degree mode or use Mathway and change answer to degrees.

Correct: cot(x)

Law of Cosines

cos A = (b² + c² – a²)/(2bc)

cos A = (43² + 50² – 29²)/(2 × 43 × 50)

cos A = 0.8158

cos⁻¹(0.8158) = 35.3°

Just do this ↓

∠A = cos⁻¹((b² + c² – a²)/(2 × 43 × 50))

∠A = 35.3°

Solution

Use Sum and Diffrence Identities:

sin(A + B) = sinAcosB + cosAsinB

sin(450°)cos(300°) + cos(450°)sin(300°) = -1/2

Mathway (Trigonometry)

Find the Exact Value

Incorrect: None of the listed answers are correct

Correct: 30°, 150°, 210°, 330°

Solution

sin(x) = √(1 – 3sin²(x))

sin²(x) = 1 – 3sin²(x)

sin²(x) + 3sin²(x) – 1

4sin²(x) – 1

(2sin(x) + 1)(2sin(x) – 1)

sin(x) = -1/2 and 1/2

x = -30°, 30°, 150°, 210°

Add 360° to -30°

-30° + 360° = 330°

x = 30°, 150°, 210°, 330°

Or graph

y = sin²(x)

y = 1 – 3sin²(x)

Intersections:

30°, 150°, 210°, 330°

Correct: 51.8

Solution

h = 54 + 25 sin(230t)

h = 54 + 25 sin(230 × 5.5)

h = 51.8

Incorrect: None of the listed answers are correct

Correct: 30°, 150°, 210°, 330°

Solution

sin(x) = √(1 – 3sin²(x))

sin²(x) = 1 – 3sin²(x)

sin²(x) + 3sin²(x) – 1

4sin²(x) – 1

(2sin(x) + 1)(2sin(x) – 1)

sin(x) = -1/2 and 1/2

x = -30°, 30°, 150°, 210°

Add 360° to -30°

-30° + 360° = 330°

x = 30°, 150°, 210°, 330°

Or graph

y = sin²(x)

y = 1 – 3sin²(x)

Intersections:

30°, 150°, 210°, 330°

Solution

F = 32°, G = 55° , E = ?°

f = 10, g = ?, e = ?

32° + 55° + ∠E = 180°

∠E = 93°

Use Law of Sines

10/sin(32) = g/sin(55)

g = 10sin(55)/sin(32)

g = 15.5

10/sin(32) = e/sin(93)

e = 10sin(93)/sin(32)

e = 18.8

Correct: 10/9, -(4√14)/9

Correct: 1 + cos(x) or 1 + cosθ

Mathway

Rationalize the Denominator

Mathway

Verify the Identity+

Mathway (Trigonometry)

Find the Exact Value

Solution

h = 12.5 + 7.5 sin(850t)

h = 12.5 + 7.5 sin(850 × 2.5)

h = 8.2

Solution

Use Law of Cosines

cos A = (b² + c² – a²)/2bc

cos A = (13² + 15² – 11²)/(2 × 13 ×15)

cos A = 7/10

cos⁻¹(7/10) = 45.6°

cos B = (c² + a² – b²)/2ca

cos B = (15² + 11² – 13²)/(2 × 15 × 11)

cos B = 59/110

cos⁻¹(59/110) = 57.6°

cos C = (a² + b² – c²)/2ab

cos C = (11² + 13² – 15²)/(2 × 11 × 13)

cos C = 5/22

cos⁻¹(5/22) = 76.9°

Use Mathway to “Verify the Identity”

csc(2x) + cot(2x) = cot(x)

Mathway

Verify the Identity

I got

√(12x – 16)/9 → √(4(3x – 4))/9

But the book has

√(6x – 8)/9 → √(2(3x -4))/9