See the attachment below!!

This law applies to all triangles, including right triangles. Basically, the law of sines is saying that the ratio between each side length and the sine of its opposite angle is the same for each angle/side combination in the triangle.

Since question three gives us two angle measures, we can use law of sines here.

Start out with the ratio, plug in your values. In this case you would use sinB, where b=7 and sinB= sin(44°), to find the remaining values. Be careful! This is usually where people mess up most frequently: they forget to put "sin(" in the denominator.

Let's solve: [tex]\frac{7}{sin(44)} =\frac{x}{sin(53)}[/tex]

We have a proportion, so cross multiply: [tex]7*sin(53) = x*sin(44)[/tex]

Solve for x accordingly: [tex]\frac{7*sin(53)}{sin(44)} = x[/tex]

Use your calculator, make sure it's in degrees, and always close the parenthesis.

You'll find that x is approximately 8.04.

If you need to find the other side and angle values, do this process again to find a and A, you can use either [tex]\frac{a}{sinA} =\frac{b}{sinB}[/tex] or [tex]\frac{a}{sinA} =\frac{c}{sinC}[/tex] since you know that values of both. You'll find that:

a ≈ 10
A = 83°

Law of Cosines states: c²=a² + b²- (2ab cosC)

Look carefully: Each of the three sides is used in the Law of Cosines, but only one angle is used. If you don't know angle C, the Law of Cosines can easily be rewritten as:

a² = b² +c² - (2bc cosA)
b² = a²+c² - (2ac cosB)

When we have all three sides, and we need to find the missing angle, we can rearrange the formula like this:

[tex]\frac{a^{2} -b^{2}-c^{2} }{-2bc}[/tex]=cosA

Again, you can rewrite the equation to find the correct angle.

Let's solve 4 since it meets the criteria for solving with the Law of Cosines.

side a = 5 cm

side b = 3 cm

side c = 5 cm

Remember: We're looking for θ, which is opposite of side b, so θ=B

Our equation will be [tex]\frac{b^{2} -a^{2}-c^{2} }{-2ac}[/tex]=cosB

Plug in the values:

(3)²-(5)²-(5)²/-2(5)(5) = cosθ

Simplify:

9-25-25/-50 = -41/-50 = 41/50 = cosθ

Isolate the variable, θ, by using the inverse of cos, arccos.

arccos41/50 = θ

Plug that into your calculator and you'll get θ=34.9152062474

Round that to 34.92 and don't forget the °!