Math Snap

STEP 1

What is this asking?
We're given two sides of a triangle ($a = 3$ and $c = 4$) and an angle ($B = 75^\circ$).
We need to find the length of the third side ($b$) and the measures of the other two angles ($A$ and $C$).
Watch out!
Remember, the angles in a triangle always add up to $180^\circ$!
Also, make sure your calculator is in degree mode, not radians!

STEP 2

1. Find side $b$
2. Find angle $C$
3. Find angle $A$

STEP 3

We're given two sides and the angle between them, so let's use the Law of Cosines to find the third side, $b$.
The Law of Cosines says: $b^2 = a^2 + c^2 - 2ac \cdot \cos(B)$.

STEP 4

Plug in our known values: $b^2 = 3^2 + 4^2 - 2 \cdot 3 \cdot 4 \cdot \cos(75^\circ)$.

STEP 5

Calculate $3^2 = 9$ and $4^2 = 16$.
Also, $2 \cdot 3 \cdot 4 = 24$.
So, $b^2 = 9 + 16 - 24 \cdot \cos(75^\circ)$.

STEP 6

Now, $\cos(75^\circ) \approx 0.2588$.
So, $b^2 \approx 25 - 24 \cdot 0.2588 \approx 25 - 6.2112 \approx 18.7888$.

STEP 7

To find $b$, we take the square root: $b \approx \sqrt{18.7888} \approx 4.33$.
So, $b \approx 4.33$.

STEP 8

Now that we have all three sides, we can use the Law of Sines to find angle $C$.
The Law of Sines says: $\frac{\sin(C)}{c} = \frac{\sin(B)}{b}$.

STEP 9

Plug in our values: $\frac{\sin(C)}{4} = \frac{\sin(75^\circ)}{4.33}$.

STEP 10

Multiply both sides by $4$ to isolate $\sin(C)$: $\sin(C) = 4 \cdot \frac{\sin(75^\circ)}{4.33} \approx 4 \cdot \frac{0.9659}{4.33} \approx 0.89$.

STEP 11

To find $C$, we take the inverse sine: $C = \arcsin(0.89) \approx 62.9^\circ$.

STEP 12

We know that the sum of the angles in a triangle is $180^\circ$.
So, $A + B + C = 180^\circ$.

STEP 13

Plug in the values we know: $A + 75^\circ + 62.9^\circ = 180^\circ$.

STEP 14

Combine the known angles: $A + 137.9^\circ = 180^\circ$.

STEP 15

Subtract $137.9^\circ$ from both sides to find $A$: $A = 180^\circ - 137.9^\circ = 42.1^\circ$.

$b \approx 4.33$, $C \approx 62.9^\circ$, and $A \approx 42.1^\circ$.