Question

Use the Law of Sines to find the indicated side x. (Assume a = 400. Round your answer to two decimal places.)

Answer 1

The measure of side length x in the triangle is approximately 339.11 units.

The figure in the image is a triangle.

To determine the missing side length, we use the sine rule.

It states that:

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

From the figure:

Angle A = 98.4 degrees

Angle B = 24.6 degrees

Angle C =?

Side a = 400

Side c = x

First, we find the measure of angle C:

A + B + C = 180

98.4 + 24.6 + C = 180

123 + C = 180

C = 180 - 123

C = 57 degrees

Now, we use the sine rule to solve for side length c:

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

Plug in the values:

400/sin(98.4 ) = x/sin( 57)

sin( 57) × 400 = sin(98.4 ) × x

x = ( sin( 57) × 400 ) / sin(98.4 )

x = 339.11 units

Therefore, the value of x is 339.11.

Answer 2

The law of sines is given by:

a/sinA = b/sinB = c/sinC

Take into account that in the given problem you need to know what is the measure of angle C, to be able to use the law of sines.

Consider that the sum of the interioiro angles of a triangle is 180°. Then, you have:

m∠C + 98.4° + 24.6° = 180°

m∠C + 123° = 180°

m∠C = 180° - 123°

m∠C = 57°

Next, use the law of sines with sides a and x, angle A and C:

a/sinA = x/sinC solve for x

(a/sinA)(sinC) = x

x = (a/sinA)(sinC) replace the values of known parameters (a = 400)

x = (400/sin98.4°)(sin57°)

x = 339.106

Hence, the length of side x is x = 339.106