Question

Law of sines: StartFraction sine (uppercase A) Over a EndFraction = StartFraction sine (uppercase B) Over b EndFraction = StartFraction sine (uppercase C) Over c EndFraction In ΔABC, c = 5. 4, a = 3. 3, and measure of angle A = 20 degrees. What are the possible approximate lengths of b? Use the law of sines to find the answer. 2. 0 units and 4. 6 units 2. 1 units and 8. 7 units 2. 3 units and 7. 8 units 2. 6 units and 6. 6 units.

Answer 1

The length 2.3 and 7.8 units

We have given that ΔABC, c = 5. 4, a = 3. 3, and measure of angle A = 20 degrees.

What is the formula for low of cosine?

The Law of Cosines, which tells us

$a^{2}=b^{2} +c^{2} -2bccosA$

giving us a quadratic equation for b we can solve.  But let's do it with the Law of Sines as asked.

$\frac{sinA}{a}=\frac{sinB}{b}=\frac{sinC}{c}$

We have c,a,A so the Law of Sines gives us sin C

$sinC=\frac{csina}{a} =\frac{5.4sin20}{3.3} =0.5597$

There are two possible triangle angles with this sine, supplementary angles, one acute, one obtuse

$C_{a}=arcsin(0.5597)=34.033^{0}$

$C_{0}=180-C_{a} =145.96^0$

Both of these make a valid triangle with A=20°.   They give respective B's:

$B_{a}=180-A-C_{a} =125.96^0$

$B_{0}=180-C_{0} =14.033^0$

So we get two possibilities for b:

$b=\frac{asinB}{sinA} \\\\b_{a} =\frac{3.3sin125.96}{sin 20} \\\\b_{a}=7.8\\\\ b_0=2.3$

2.3 units and 7.8 units

Therefore we get the length 2.3 and 7.8 units

To learn more about the low of sine and cosine visit:

brainly.com/question/4372174

Answer: 2.3 units and 7.8 units

Let's check it with the Law of Cosines:

There's a shortcut for the quadratic formula when the middle term is 'even.'

Looks good.