Math / Geometry

QuestionFind $c$ if $a=2.14 \mathrm{mi}, b=3.99 \mathrm{mi}$ and $\angle C=40.9$ degrees. $c=$ $\square$ mi ;
Assume $\angle A$ is opposite side $a, \angle B$ is opposite side $b$ , and $\angle C$ is opposite side $c$ . Question Help: Video

Studdy Solution

STEP 1

What is this asking? We're given two sides of a triangle and the angle between them (SAS triangle), and we need to find the length of the third side. Watch out! Make sure your calculator is in degree mode, not radians!
Also, remember that the Law of Cosines has a plus sign, not a minus sign, when the angle is between the two given sides.

STEP 2

1. Apply the Law of Cosines
2. Calculate the result

STEP 3

We're given two sides and the angle between them.
This is a classic setup for the Law of Cosines!
It's like having two pieces of a puzzle and the angle that connects them.
The Law of Cosines helps us find the missing piece!

STEP 4

The Law of Cosines states: $c^2 = a^2 + b^2 - 2 \cdot a \cdot b \cdot \cos(C)$ .
Here, $c$ is the side we want to find, $a$ and $b$ are the given sides, and $C$ is the angle between them.

STEP 5

Let's plug in our values: $a = \mathbf{2.14}$ mi, $b = \mathbf{3.99}$ mi, and $C = \mathbf{40.9}$ degrees.
So, we have $c^2 = (\mathbf{2.14})^2 + (\mathbf{3.99})^2 - 2 \cdot (\mathbf{2.14}) \cdot (\mathbf{3.99}) \cdot \cos(\mathbf{40.9})$ .

STEP 6

$(\mathbf{2.14})^2 = \mathbf{4.5796}$ and $(\mathbf{3.99})^2 = \mathbf{15.9201}$ .
So, our equation becomes $c^2 = \mathbf{4.5796} + \mathbf{15.9201} - 2 \cdot (\mathbf{2.14}) \cdot (\mathbf{3.99}) \cdot \cos(\mathbf{40.9})$ .

STEP 7

$\cos(\mathbf{40.9}) \approx \mathbf{0.7558}$ .
Now, our equation is $c^2 = \mathbf{4.5796} + \mathbf{15.9201} - 2 \cdot (\mathbf{2.14}) \cdot (\mathbf{3.99}) \cdot \mathbf{0.7558}$ .

STEP 8

$2 \cdot (\mathbf{2.14}) \cdot (\mathbf{3.99}) \cdot \mathbf{0.7558} \approx \mathbf{12.9045}$ .
So, $c^2 = \mathbf{4.5796} + \mathbf{15.9201} - \mathbf{12.9045}$ .

STEP 9

$\mathbf{4.5796} + \mathbf{15.9201} = \mathbf{20.4997}$ .
Then, $\mathbf{20.4997} - \mathbf{12.9045} = \mathbf{7.5952}$ .
Therefore, $c^2 = \mathbf{7.5952}$ .

STEP 10

Finally, we take the square root of both sides to get $c = \sqrt{\mathbf{7.5952}} \approx \mathbf{2.756}$ mi.
Remember, we're looking for a length, so we only consider the positive square root.

STEP 11

$c \approx \mathbf{2.756}$ mi.

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QuestionFind $c$ if $a=2.14 \mathrm{mi}, b=3.99 \mathrm{mi}$ and $\angle C=40.9$ degrees. $c=$ $\square$ mi ;
Assume $\angle A$ is opposite side $a, \angle B$ is opposite side $b$ , and $\angle C$ is opposite side $c$ . Question Help: Video

Studdy Solution

STEP 1

STEP 2

1. Apply the Law of Cosines
2. Calculate the result

STEP 3

We're given two sides and the angle between them.
This is a classic setup for the Law of Cosines!
It's like having two pieces of a puzzle and the angle that connects them.
The Law of Cosines helps us find the missing piece!

STEP 4

The Law of Cosines states: $c^2 = a^2 + b^2 - 2 \cdot a \cdot b \cdot \cos(C)$ .
Here, $c$ is the side we want to find, $a$ and $b$ are the given sides, and $C$ is the angle between them.

STEP 5

Let's plug in our values: $a = \mathbf{2.14}$ mi, $b = \mathbf{3.99}$ mi, and $C = \mathbf{40.9}$ degrees.
So, we have $c^2 = (\mathbf{2.14})^2 + (\mathbf{3.99})^2 - 2 \cdot (\mathbf{2.14}) \cdot (\mathbf{3.99}) \cdot \cos(\mathbf{40.9})$ .

STEP 6

$(\mathbf{2.14})^2 = \mathbf{4.5796}$ and $(\mathbf{3.99})^2 = \mathbf{15.9201}$ .
So, our equation becomes $c^2 = \mathbf{4.5796} + \mathbf{15.9201} - 2 \cdot (\mathbf{2.14}) \cdot (\mathbf{3.99}) \cdot \cos(\mathbf{40.9})$ .

STEP 7

$\cos(\mathbf{40.9}) \approx \mathbf{0.7558}$ .
Now, our equation is $c^2 = \mathbf{4.5796} + \mathbf{15.9201} - 2 \cdot (\mathbf{2.14}) \cdot (\mathbf{3.99}) \cdot \mathbf{0.7558}$ .

STEP 8

$2 \cdot (\mathbf{2.14}) \cdot (\mathbf{3.99}) \cdot \mathbf{0.7558} \approx \mathbf{12.9045}$ .
So, $c^2 = \mathbf{4.5796} + \mathbf{15.9201} - \mathbf{12.9045}$ .

STEP 9

$\mathbf{4.5796} + \mathbf{15.9201} = \mathbf{20.4997}$ .
Then, $\mathbf{20.4997} - \mathbf{12.9045} = \mathbf{7.5952}$ .
Therefore, $c^2 = \mathbf{7.5952}$ .

STEP 10

Finally, we take the square root of both sides to get $c = \sqrt{\mathbf{7.5952}} \approx \mathbf{2.756}$ mi.
Remember, we're looking for a length, so we only consider the positive square root.

STEP 11

$c \approx \mathbf{2.756}$ mi.

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Studdy Solution

STEP 1

STEP 2

1. Apply the Law of Cosines2. Calculate the result

STEP 3

We're given two sides and the angle *between* them.This is a classic setup for the Law of Cosines!It's like having two pieces of a puzzle and the angle that connects them.The Law of Cosines helps us find the missing piece!

STEP 4

The Law of Cosines states: c2=a2+b2−2⋅a⋅b⋅cos⁡(C)c^2 = a^2 + b^2 - 2 \cdot a \cdot b \cdot \cos(C)c2=a2+b2−2⋅a⋅b⋅cos(C).Here, ccc is the side we want to find, aaa and bbb are the given sides, and CCC is the angle between them.

STEP 5

STEP 6

STEP 7

STEP 8

2⋅(2.14)⋅(3.99)⋅0.7558≈12.90452 \cdot (\mathbf{2.14}) \cdot (\mathbf{3.99}) \cdot \mathbf{0.7558} \approx \mathbf{12.9045}2⋅(2.14)⋅(3.99)⋅0.7558≈12.9045.So, c2=4.5796+15.9201−12.9045c^2 = \mathbf{4.5796} + \mathbf{15.9201} - \mathbf{12.9045}c2=4.5796+15.9201−12.9045.

STEP 9

4.5796+15.9201=20.4997\mathbf{4.5796} + \mathbf{15.9201} = \mathbf{20.4997}4.5796+15.9201=20.4997.Then, 20.4997−12.9045=7.5952\mathbf{20.4997} - \mathbf{12.9045} = \mathbf{7.5952}20.4997−12.9045=7.5952.Therefore, c2=7.5952c^2 = \mathbf{7.5952}c2=7.5952.

STEP 10

Finally, we take the square root of both sides to get c=7.5952≈2.756c = \sqrt{\mathbf{7.5952}} \approx \mathbf{2.756}c=7.5952​≈2.756 mi.Remember, we're looking for a length, so we only consider the positive square root.

STEP 11

c≈2.756c \approx \mathbf{2.756}c≈2.756 mi.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Apply the Law of Cosines
2. Calculate the result

We're given two sides and the angle between them.
This is a classic setup for the Law of Cosines!
It's like having two pieces of a puzzle and the angle that connects them.
The Law of Cosines helps us find the missing piece!

The Law of Cosines states: $c^2 = a^2 + b^2 - 2 \cdot a \cdot b \cdot \cos(C)$ .
Here, $c$ is the side we want to find, $a$ and $b$ are the given sides, and $C$ is the angle between them.

$2 \cdot (\mathbf{2.14}) \cdot (\mathbf{3.99}) \cdot \mathbf{0.7558} \approx \mathbf{12.9045}$ .
So, $c^2 = \mathbf{4.5796} + \mathbf{15.9201} - \mathbf{12.9045}$ .

$\mathbf{4.5796} + \mathbf{15.9201} = \mathbf{20.4997}$ .
Then, $\mathbf{20.4997} - \mathbf{12.9045} = \mathbf{7.5952}$ .
Therefore, $c^2 = \mathbf{7.5952}$ .

Finally, we take the square root of both sides to get $c = \sqrt{\mathbf{7.5952}} \approx \mathbf{2.756}$ mi.
Remember, we're looking for a length, so we only consider the positive square root.

$c \approx \mathbf{2.756}$ mi.