could someone please help.

given each set of data for ∆ABC:
a) solve for b given ∠B=88 degrees,a=31.5m,c=25.6cm
b) solve for a given ∠B=66 degrees,∠C=28 degrees,c=11.6cm
i) state whether the sine law or cosine law is required to solve the unknown value.
ii) use the law you selected in i) to solve for the unknown value, correct to one decimal place.

Use cosines law for SSS and SAS

use sine law for others

a) cosine law
b) sine law

thank you

To solve these problems, we can use the sine law or cosine law. Let's determine which law we need for each question.

a) To solve for side b, given ∠B=88 degrees, a=31.5m, and c=25.6cm:
i) In this case, we need to use the sine law because we are given an angle and the opposite side.

b) To solve for side a, given ∠B=66 degrees, ∠C=28 degrees, and c=11.6cm:
i) In this case, we need to use the cosine law because we are given two angles and one side.

Now, let's use the appropriate law to solve for the unknown values.

a) Sine law is required:
The sine law states that for any triangle ABC,
a/sin(A) = b/sin(B) = c/sin(C)

Given: ∠B = 88 degrees, a = 31.5m, c = 25.6cm
We want to solve for b.

From the sine law, we can set up the proportion:
b/sin(B) = c/sin(C)

Plugging in the given values, we get:
b/sin(88) = 25.6/sin(C)

To find sin(C), we can use the fact that the sum of angles in a triangle is 180 degrees:
∠A + ∠B + ∠C = 180
∠A + 88 + ∠C = 180
∠A + ∠C = 92

Since we know that ∠A + ∠C = 92 and ∠C = 180 - ∠A - ∠B, we can substitute ∠C in the proportion:
b/sin(88) = 25.6/sin(180 - ∠A - ∠B)

By substituting the given values, we get:
b/sin(88) = 25.6/sin(180 - ∠A - 88)

Now, we can solve for b by cross-multiplying and isolating b:
b = (sin(88) * 25.6) / sin(180 - ∠A - 88)

To get the answer, plug the values of the angles into the equation and calculate b, rounded to one decimal place.

b) Cosine law is required:
The cosine law states that for any triangle ABC,
c^2 = a^2 + b^2 - 2ab * cos(C)

Given: ∠B = 66 degrees, ∠C = 28 degrees, c = 11.6cm
We want to solve for a.

From the cosine law, we can set up the equation:
c^2 = a^2 + b^2 - 2ab * cos(C)

Plugging in the given values, we get:
11.6^2 = a^2 + b^2 - 2ab * cos(28)

To solve for a, we need to rearrange the equation:
a^2 = 11.6^2 - b^2 + 2ab * cos(28)

Now, we can solve for a by substituting the values for b and calculating a, rounded to one decimal place.

Note: It's important to convert all angle measurements to radians when using trigonometric functions in calculations.