Math / Geometry

QuestionIn the following exercise, two sides and an angle are given. First determine whether the information results in no triangle, one triangle, or two triangles. Solve the resulting triangle. $a=9.3, b=7.2 \text {, and } A=36^{\circ}$
Determine the value of $\sin B$ . $\sin B=27.07$ (Round to four decimal places as needed.)

Studdy Solution

STEP 1

What is this asking? We need to figure out if the given sides and angle can form a triangle, and if so, find the value of $\sin B$ . Watch out! Be careful with the ambiguous case of the Law of Sines; sometimes there can be two triangles or none at all!

STEP 2

1. Check if a triangle is possible
2. Use the Law of Sines
3. Solve for $\sin B$

STEP 3

Alright, let's start by checking if a triangle can be formed with the given information.
We have side $a = 9.3$ , side $b = 7.2$ , and angle $A = 36^\circ$ .
This is a classic SSA (Side-Side-Angle) situation, which can be a bit tricky!

STEP 4

To determine if a triangle is possible, we need to check if the side opposite the given angle $A$ (which is $a$ ) is longer than the height of the triangle when side $b$ is the base.
The height $h$ can be calculated using the formula: $h = b \cdot \sin A$

STEP 5

Plug in the values: $h = 7.2 \cdot \sin(36^\circ)$

STEP 6

Calculate $\sin(36^\circ)$ using a calculator: $\sin(36^\circ) \approx 0.5878$

STEP 7

Now, calculate $h$ : $h = 7.2 \cdot 0.5878 \approx 4.23$

STEP 8

Since $a = 9.3$ is greater than $h = 4.23$ , a triangle is possible.
In fact, since $a > b$ , there is only one triangle possible.

STEP 9

Now, let's use the Law of Sines to find $\sin B$ .
The Law of Sines states: $\frac{a}{\sin A} = \frac{b}{\sin B}$

STEP 10

Rearrange the formula to solve for $\sin B$ : $\sin B = \frac{b \cdot \sin A}{a}$

STEP 11

Plug in the known values: $\sin B = \frac{7.2 \cdot \sin(36^\circ)}{9.3}$

STEP 12

Substitute the value of $\sin(36^\circ)$ we calculated earlier: $\sin B = \frac{7.2 \cdot 0.5878}{9.3}$

STEP 13

Calculate the numerator: $7.2 \cdot 0.5878 \approx 4.23$

STEP 14

Now, divide by $9.3$ : $\sin B = \frac{4.23}{9.3} \approx 0.4548$

STEP 15

The value of $\sin B$ is approximately 0.4548.

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Studdy Solution

STEP 1

What is this asking? We need to figure out if the given sides and angle can form a triangle, and if so, find the value of $\sin B$ . Watch out! Be careful with the ambiguous case of the Law of Sines; sometimes there can be two triangles or none at all!

STEP 2

1. Check if a triangle is possible
2. Use the Law of Sines
3. Solve for $\sin B$

STEP 3

Alright, let's start by checking if a triangle can be formed with the given information.
We have side $a = 9.3$ , side $b = 7.2$ , and angle $A = 36^\circ$ .
This is a classic SSA (Side-Side-Angle) situation, which can be a bit tricky!

STEP 4

To determine if a triangle is possible, we need to check if the side opposite the given angle $A$ (which is $a$ ) is longer than the height of the triangle when side $b$ is the base.
The height $h$ can be calculated using the formula: $h = b \cdot \sin A$

STEP 5

Plug in the values: $h = 7.2 \cdot \sin(36^\circ)$

STEP 6

Calculate $\sin(36^\circ)$ using a calculator: $\sin(36^\circ) \approx 0.5878$

STEP 7

Now, calculate $h$ : $h = 7.2 \cdot 0.5878 \approx 4.23$

STEP 8

Since $a = 9.3$ is greater than $h = 4.23$ , a triangle is possible.
In fact, since $a > b$ , there is only one triangle possible.

STEP 9

Now, let's use the Law of Sines to find $\sin B$ .
The Law of Sines states: $\frac{a}{\sin A} = \frac{b}{\sin B}$

STEP 10

Rearrange the formula to solve for $\sin B$ : $\sin B = \frac{b \cdot \sin A}{a}$

STEP 11

Plug in the known values: $\sin B = \frac{7.2 \cdot \sin(36^\circ)}{9.3}$

STEP 12

Substitute the value of $\sin(36^\circ)$ we calculated earlier: $\sin B = \frac{7.2 \cdot 0.5878}{9.3}$

STEP 13

Calculate the numerator: $7.2 \cdot 0.5878 \approx 4.23$

STEP 14

Now, divide by $9.3$ : $\sin B = \frac{4.23}{9.3} \approx 0.4548$

STEP 15

The value of $\sin B$ is approximately 0.4548.

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Studdy Solution

STEP 1

What is this asking? We need to figure out if the given sides and angle can form a triangle, and if so, find the value of sin⁡B\sin BsinB. Watch out! Be careful with the ambiguous case of the Law of Sines; sometimes there can be two triangles or none at all!

STEP 2

1. Check if a triangle is possible2. Use the Law of Sines3. Solve for sin⁡B\sin BsinB

STEP 3

Alright, let's start by checking if a triangle can be formed with the given information.We have side a=9.3a = 9.3a=9.3, side b=7.2b = 7.2b=7.2, and angle A=36∘A = 36^\circA=36∘.This is a classic **SSA** (Side-Side-Angle) situation, which can be a bit tricky!

STEP 4

To determine if a triangle is possible, we need to check if the side opposite the given angle AAA (which is aaa) is longer than the height of the triangle when side bbb is the base.The height hhh can be calculated using the formula: h=b⋅sin⁡Ah = b \cdot \sin Ah=b⋅sinA

STEP 5

Plug in the values: h=7.2⋅sin⁡(36∘)h = 7.2 \cdot \sin(36^\circ)h=7.2⋅sin(36∘)

STEP 6

Calculate sin⁡(36∘)\sin(36^\circ)sin(36∘) using a calculator: sin⁡(36∘)≈0.5878\sin(36^\circ) \approx 0.5878sin(36∘)≈0.5878

STEP 7

Now, calculate hhh: h=7.2⋅0.5878≈4.23h = 7.2 \cdot 0.5878 \approx 4.23h=7.2⋅0.5878≈4.23

STEP 8

Since a=9.3a = 9.3a=9.3 is greater than h=4.23h = 4.23h=4.23, a triangle is possible.In fact, since a>ba > ba>b, there is only one triangle possible.

STEP 9

Now, let's use the **Law of Sines** to find sin⁡B\sin BsinB.The Law of Sines states: asin⁡A=bsin⁡B\frac{a}{\sin A} = \frac{b}{\sin B}sinAa​=sinBb​

STEP 10

Rearrange the formula to solve for sin⁡B\sin BsinB: sin⁡B=b⋅sin⁡Aa\sin B = \frac{b \cdot \sin A}{a}sinB=ab⋅sinA​

STEP 11

Plug in the known values: sin⁡B=7.2⋅sin⁡(36∘)9.3\sin B = \frac{7.2 \cdot \sin(36^\circ)}{9.3}sinB=9.37.2⋅sin(36∘)​

STEP 12

Substitute the value of sin⁡(36∘)\sin(36^\circ)sin(36∘) we calculated earlier: sin⁡B=7.2⋅0.58789.3\sin B = \frac{7.2 \cdot 0.5878}{9.3}sinB=9.37.2⋅0.5878​

STEP 13

Calculate the numerator: 7.2⋅0.5878≈4.237.2 \cdot 0.5878 \approx 4.237.2⋅0.5878≈4.23

STEP 14

Now, divide by 9.39.39.3: sin⁡B=4.239.3≈0.4548\sin B = \frac{4.23}{9.3} \approx 0.4548sinB=9.34.23​≈0.4548

STEP 15

The value of sin⁡B\sin BsinB is approximately **0.4548**.

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What is this asking? We need to figure out if the given sides and angle can form a triangle, and if so, find the value of $\sin B$ . Watch out! Be careful with the ambiguous case of the Law of Sines; sometimes there can be two triangles or none at all!

1. Check if a triangle is possible
2. Use the Law of Sines
3. Solve for $\sin B$

Alright, let's start by checking if a triangle can be formed with the given information.
We have side $a = 9.3$ , side $b = 7.2$ , and angle $A = 36^\circ$ .
This is a classic SSA (Side-Side-Angle) situation, which can be a bit tricky!

To determine if a triangle is possible, we need to check if the side opposite the given angle $A$ (which is $a$ ) is longer than the height of the triangle when side $b$ is the base.
The height $h$ can be calculated using the formula: $h = b \cdot \sin A$

Plug in the values: $h = 7.2 \cdot \sin(36^\circ)$

Calculate $\sin(36^\circ)$ using a calculator: $\sin(36^\circ) \approx 0.5878$

Now, calculate $h$ : $h = 7.2 \cdot 0.5878 \approx 4.23$

Since $a = 9.3$ is greater than $h = 4.23$ , a triangle is possible.
In fact, since $a > b$ , there is only one triangle possible.

Now, let's use the Law of Sines to find $\sin B$ .
The Law of Sines states: $\frac{a}{\sin A} = \frac{b}{\sin B}$

Rearrange the formula to solve for $\sin B$ : $\sin B = \frac{b \cdot \sin A}{a}$

Plug in the known values: $\sin B = \frac{7.2 \cdot \sin(36^\circ)}{9.3}$

Substitute the value of $\sin(36^\circ)$ we calculated earlier: $\sin B = \frac{7.2 \cdot 0.5878}{9.3}$

Calculate the numerator: $7.2 \cdot 0.5878 \approx 4.23$

Now, divide by $9.3$ : $\sin B = \frac{4.23}{9.3} \approx 0.4548$

The value of $\sin B$ is approximately 0.4548.