Question

In AABC, A = 42°, a = 16 and c = 20. Which of these statements best describes angle C?

Answer 1

The statement that best describes the angle C is that C must be acute

Which of the statements best describes angle C?

From the question, we have the following parameters that can be used in our computation:

A = 42°, a = 16 and c = 20

The measure of the angle C is calculated using

c/sin(C) = a/sin(A)

Substitute the known values into the equation

20/sin(C) = 16/sin(42)

This gives

20/sin(C) = 23.91

So, we have

sin(C) = 20/23.91

sin(C) = 0.8365

Take the arc sin of both sides

C = 56.8

Hence, the statements best describes angle C is that C must be acute

Answer 2

Hello there. To solve this question, we'll need to remember how to use the law of sines.

Say we have the following triangle:

We know that a = 16, c = 20 and A = 42º

Assuming that this is the right orientation for this triangle, I mean, the angle A is opposite to the side a and the angle C is opposite to the side c, we can make use of the law of sines:

We'll have:

16/sin(42º) = 20/sin(C)

Divide both sides of the equation by a factor of 4

4/sin(42º) = 20/sin(C)

Considering we have a triangle, none of the angles can be zero.

Such that sin(any angle) greater than zero.

We'll make the following inequality, to make our lifes easier to deal with the sines:

sin is an injective function, such that if you have two angles a1 and a2, with a2 > a1, then sin(a2) > sin(a1)

Which means that sin(45º) > sin(42º)

Inverting the inequality, we'll have:

1/sin(42º) > 1/sin(45º)

Multiply both sides by 4

4/sin(42º) > 4/sin(45º)

But we know that 4/sin(42º) = 5/sin(C), such that:

5/sin(C) > 4/sin(45º)

Divide both sides of the equation by a factor of 5

1/sin(C) > (4/5)/sin(45º)

Inverting the inequality once again, we'll have

sin(C) < (5/4) * sin(45º)

Knowing that sin(45º) = 1/sqrt(2) approx. 0.707, we get

0 < sin(C) < 0.883

In which 0.883 < 1, such that

0 < sin(C) < 1

Then

0 < C < 90º

C is an acute angle.