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inna [77]
3 years ago
9

Given : cos(θ)= -4/5 | sin x = -12/13 Evaluate: cos(θ + x)

Mathematics
1 answer:
artcher [175]3 years ago
8 0

cos (θ + x) is 16/65

Step-by-step explanation:

  • Step 1:

Given cos θ = -4/5, find sin θ.

cos θ = adjacent side/hypotenuse = -4/5

The opposite side can be found using Pythagoras Theorem.

Hypotenuse² = Opposite Side² + Adjacent Side²

⇒ Opposite side² = Hypotenuse² - Adjacent Side²

                             = 5² - (-4)² = 25 - 16 = 9

∴ Opposite Side = 3

⇒ sin θ = opposite side/hypotenuse = 3/5

  • Step 2:

Given sin x = -12/13, find cos x.

sin x = opposite side/hypotenuse = -12/13

The adjacent side can be found using Pythagoras Theorem.

Hypotenuse² = Opposite Side² + Adjacent Side²

⇒ Adjacent side² = Hypotenuse² - Adjacent Side²

                             = 13² - (-12)² = 169 - 144 = 25

∴ Adjacent Side = 5

⇒ cos x = adjacent side/hypotenuse = 5/13

  • Step 3:

Find cos (θ + x).

cos (θ + x) = cos θ cos x - sin θ sin x

                 = -4/5 × 5/13 - 3/5 × -12/13

                 = -20/65 + 36/65 = 16/65

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Recall the scenario about Eric's weekly wages in the lesson practice section. Eric's boss have been very impressed with his work
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Answer:  

1)\quad f(x)=\bigg\{\begin{array}{ll}12x&0\leq x

2) D: x = [0, 24]

3) R: y = [0, 384]

4) see graph

<u>Step-by-step explanation:</u>

Eric's regular wage is $12 per hour for all hours less than 9 hours.

The minimum number of hours Eric can work each day is 0.

f(x) = 12x    for   0 ≤ x < 9

Eric's overtime wage is $18 per hour for 9 hours and greater.

The maximum number of hours Eric can work each day is 24 (because there are only 24 hours in a day).

f(x) = 18(x - 8) + 12(8)

    = 18x - 144 + 96

    = 18x - 48           for 9 ≤ x ≤ 24

The daily wage where x represents the number of hours worked can be displayed in function format as follows:

f(x)=\bigg\{\begin{array}{ll}12x&0\leq x

2) Domain represents the x-values (number of hours Eric can work).

The minimum hours he can work in one day is 0 and the maximum he can work in one day is 24.

D:  0 ≤ x ≤ 24        →        D: x = [0, 24]

3) Range represents the y-values (wage Eric will earn).

Eric's wage depends on the number of hours he works. Use the Domain (given above) to find the wage.

The minimum hours he can work in one day is 0.

f(x) = 12x

f(0) = 12(0)

     =  0

The maximum hours he can work in one day is 24 <em>(although unlikely, it is theoretically possible).</em>

f(x) = 18x - 48

f(24) = 18(24) - 48

       = 432 - 48

       = 384

D:  0 ≤ y ≤ 384        →        D: x = [0, 384]

4) see graph.

Notice that there is an open dot at x = 9 for f(x) = 12x

and a closed dot at x = 9 for f(x) = 18x - 48

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