Math / Geometry

QuestionFind $a$ if $b=177 \mathrm{mi}, c=171 \mathrm{mi}$ and $\angle \alpha=169^{\circ}$ . $a=$ $\square$ mi ;
Assume $\angle \alpha$ is opposite side $a, \angle \beta$ is opposite side $b$ , and $\angle \gamma$ is opposite side $c$ .

Studdy Solution

STEP 1

What is this asking? We're given two sides of a triangle and the angle between them, and we need to find the length of the third side. Watch out! Make sure your calculator is in degree mode, and don't mix up the sides and angles!

STEP 2

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

STEP 3

Alright, let's do this!
We're given two sides, $b = 177$ mi and $c = 171$ mi, and the angle between them, $\alpha = 169^\circ$ .
We want to find the third side, $a$ .
The Law of Cosines is the perfect tool for this job!

STEP 4

The Law of Cosines states: $a^2 = b^2 + c^2 - 2 \cdot b \cdot c \cdot \cos(\alpha)$ This formula connects the lengths of the sides of any triangle with the cosine of one of its angles.

STEP 5

Let's plug in our known values: $a^2 = (177)^2 + (171)^2 - 2 \cdot (177) \cdot (171) \cdot \cos(169^\circ)$

STEP 6

Time to crunch some numbers!
First, let's square those side lengths: $a^2 = 31329 + 29241 - 60414 \cdot \cos(169^\circ)$

STEP 7

Now, let's evaluate the cosine: $\cos(169^\circ) \approx -0.9816$ So, $a^2 \approx 31329 + 29241 - 60414 \cdot (-0.9816)$

STEP 8

Let's multiply and then add everything up: $a^2 \approx 60570 + 59302.97$ $a^2 \approx 119872.97$

STEP 9

Almost there!
We need to find $a$ , not $a^2$ .
So, we take the square root of both sides: $a = \sqrt{119872.97}$ $a \approx 346.23$

STEP 10

The length of side $a$ is approximately $346.23$ mi.

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QuestionFind $a$ if $b=177 \mathrm{mi}, c=171 \mathrm{mi}$ and $\angle \alpha=169^{\circ}$ . $a=$ $\square$ mi ;
Assume $\angle \alpha$ is opposite side $a, \angle \beta$ is opposite side $b$ , and $\angle \gamma$ is opposite side $c$ .

Studdy Solution

STEP 1

What is this asking? We're given two sides of a triangle and the angle between them, and we need to find the length of the third side. Watch out! Make sure your calculator is in degree mode, and don't mix up the sides and angles!

STEP 2

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

STEP 3

Alright, let's do this!
We're given two sides, $b = 177$ mi and $c = 171$ mi, and the angle between them, $\alpha = 169^\circ$ .
We want to find the third side, $a$ .
The Law of Cosines is the perfect tool for this job!

STEP 4

The Law of Cosines states: $a^2 = b^2 + c^2 - 2 \cdot b \cdot c \cdot \cos(\alpha)$ This formula connects the lengths of the sides of any triangle with the cosine of one of its angles.

STEP 5

Let's plug in our known values: $a^2 = (177)^2 + (171)^2 - 2 \cdot (177) \cdot (171) \cdot \cos(169^\circ)$

STEP 6

Time to crunch some numbers!
First, let's square those side lengths: $a^2 = 31329 + 29241 - 60414 \cdot \cos(169^\circ)$

STEP 7

Now, let's evaluate the cosine: $\cos(169^\circ) \approx -0.9816$ So, $a^2 \approx 31329 + 29241 - 60414 \cdot (-0.9816)$

STEP 8

Let's multiply and then add everything up: $a^2 \approx 60570 + 59302.97$ $a^2 \approx 119872.97$

STEP 9

Almost there!
We need to find $a$ , not $a^2$ .
So, we take the square root of both sides: $a = \sqrt{119872.97}$ $a \approx 346.23$

STEP 10

The length of side $a$ is approximately $346.23$ mi.

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Studdy Solution

STEP 1

What is this asking? We're given two sides of a triangle and the angle between them, and we need to find the length of the third side. Watch out! Make sure your calculator is in degree mode, and don't mix up the sides and angles!

STEP 2

1. Apply the Law of Cosines2. Calculate the result

STEP 3

Alright, let's **do this**!We're given two sides, b=177b = 177b=177 mi and c=171c = 171c=171 mi, and the angle between them, α=169∘\alpha = 169^\circα=169∘.We want to find the third side, aaa.The **Law of Cosines** is the perfect tool for this job!

STEP 4

The Law of Cosines states: a2=b2+c2−2⋅b⋅c⋅cos⁡(α)a^2 = b^2 + c^2 - 2 \cdot b \cdot c \cdot \cos(\alpha)a2=b2+c2−2⋅b⋅c⋅cos(α) This formula connects the lengths of the sides of any triangle with the cosine of one of its angles.

STEP 5

Let's **plug in** our **known values**: a2=(177)2+(171)2−2⋅(177)⋅(171)⋅cos⁡(169∘)a^2 = (177)^2 + (171)^2 - 2 \cdot (177) \cdot (171) \cdot \cos(169^\circ)a2=(177)2+(171)2−2⋅(177)⋅(171)⋅cos(169∘)

STEP 6

Time to **crunch some numbers**!First, let's square those side lengths: a2=31329+29241−60414⋅cos⁡(169∘)a^2 = 31329 + 29241 - 60414 \cdot \cos(169^\circ)a2=31329+29241−60414⋅cos(169∘)

STEP 7

Now, let's evaluate the cosine: cos⁡(169∘)≈−0.9816\cos(169^\circ) \approx -0.9816cos(169∘)≈−0.9816 So, a2≈31329+29241−60414⋅(−0.9816)a^2 \approx 31329 + 29241 - 60414 \cdot (-0.9816)a2≈31329+29241−60414⋅(−0.9816)

STEP 8

Let's **multiply** and then **add everything up**: a2≈60570+59302.97a^2 \approx 60570 + 59302.97a2≈60570+59302.97 a2≈119872.97a^2 \approx 119872.97a2≈119872.97

STEP 9

Almost there!We need to find aaa, not a2a^2a2.So, we take the **square root** of both sides: a=119872.97a = \sqrt{119872.97}a=119872.97​ a≈346.23a \approx 346.23a≈346.23

STEP 10

The length of side aaa is approximately 346.23346.23346.23 mi.

Get Studdy

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

Alright, let's do this!
We're given two sides, $b = 177$ mi and $c = 171$ mi, and the angle between them, $\alpha = 169^\circ$ .
We want to find the third side, $a$ .
The Law of Cosines is the perfect tool for this job!

The Law of Cosines states: $a^2 = b^2 + c^2 - 2 \cdot b \cdot c \cdot \cos(\alpha)$ This formula connects the lengths of the sides of any triangle with the cosine of one of its angles.

Let's plug in our known values: $a^2 = (177)^2 + (171)^2 - 2 \cdot (177) \cdot (171) \cdot \cos(169^\circ)$

Time to crunch some numbers!
First, let's square those side lengths: $a^2 = 31329 + 29241 - 60414 \cdot \cos(169^\circ)$

Now, let's evaluate the cosine: $\cos(169^\circ) \approx -0.9816$ So, $a^2 \approx 31329 + 29241 - 60414 \cdot (-0.9816)$

Let's multiply and then add everything up: $a^2 \approx 60570 + 59302.97$ $a^2 \approx 119872.97$

Almost there!
We need to find $a$ , not $a^2$ .
So, we take the square root of both sides: $a = \sqrt{119872.97}$ $a \approx 346.23$

The length of side $a$ is approximately $346.23$ mi.