Question

review - trigonometry and periodic functions

b)
triangle def with side lengths:
de = 5.5 cm
df = 5.3 cm
ef = 5 cm
angle d = x

find the value of x.

Explanation:

Identify given triangle values

We analyze triangle \(DEF\).
The side lengths are:
\(d = 5\text{ cm}\) (opposite to vertex \(D\))
\(e = 5.3\text{ cm}\) (opposite to vertex \(E\))
\(f = 5.5\text{ cm}\) (opposite to vertex \(F\))
The target unknown is angle \(D\), labeled as \(x\).

State the Law of Cosines

To find an angle when three sides are known, we use the Law of Cosines:

$$d^2 = e^2 + f^2 - 2ef \cos(D)$$

Rearrange for the cosine term

Isolating \(\cos(D)\) gives:

$$\cos(D) = \frac{e^2 + f^2 - d^2}{2ef}$$

Substitute values and calculate

Substitute the given side lengths:

$$\cos(x) = \frac{5.3^2 + 5.5^2 - 5^2}{2(5.3)(5.5)}$$

$$\cos(x) = \frac{28.09 + 30.25 - 25}{58.3}$$

$$\cos(x) = \frac{33.34}{58.3} \approx 0.57187$$

Solve for the angle

Take the inverse cosine to find \(x\):

$$x = \arccos(0.57187) \approx 55.1^\circ$$

Answer:

\(x \approx 55.1^\circ\)