Question

if the length of a is 16 centimeters and the measures of C is 22 degrees what is the length of c round your answer to two decimal places.

Answer 1

The length of side c, given that the length of side a is 16 centimeters and the measure of angle C is 22 degrees, is approximately
`\(c \approx 17.83\)` centimeters (rounded to two decimal places).

Step-by-step explanation:

In a triangle, the relationship between the lengths of the sides and the measures of the angles is defined by trigonometric ratios. The law of sines, in particular, is applicable to this scenario.

1. Law of Sines:

According to the Law of Sines, the ratio of the length of a side to the sine of its opposite angle is constant for all sides in a triangle. Mathematically, for a triangle ABC with sides a, b, and c opposite to angles A, B, and C respectively:

`\[ (a)/(\sin A) = (b)/(\sin B) = (c)/(\sin C) \]`

2. Calculations:

Given \(a = 16\) centimeters and
`\(C = 22\)` degrees, we can rearrange the formula to solve for c:

`\[ c = (a)/(\sin C) \] Substituting the values:`

`\[ c = (16)/(\sin 22^\circ) \]`

Using a calculator to find
`\(\sin 22^\circ\)` and performing the division, we get
`\(c \approx 17.83\)` centimeters (rounded to two decimal places).

Understanding trigonometric relationships is essential for solving geometric problems involving triangles and angles.