find $m\\angle s$.\nwrite your answer as an integer or as a decimal rounded to the nearest tenth.\n$m\\angle…
find $m\\angle s$.\nwrite your answer as an integer or as a decimal rounded to the nearest tenth.\n$m\\angle s = \\square^{\\circ}$
Answer
Explanation:
Step1: Identify the given side lengths of the triangle
The side lengths are $RS = 14$, $RT = 15$, and $ST = 21$.
Step2: Apply the Law of Cosines to find angle $S$
The Law of Cosines states $RT^2 = RS^2 + ST^2 - 2(RS)(ST)\cos(S)$.
Step3: Substitute the known values into the formula
$$15^2 = 14^2 + 21^2 - 2(14)(21)\cos(S)$$
Step4: Simplify the numerical values
$$225 = 196 + 441 - 588\cos(S)$$
Step5: Isolate the cosine term
$$225 = 637 - 588\cos(S) \implies -412 = -588\cos(S)$$
Step6: Solve for $\cos(S)$
$$\cos(S) = \frac{412}{588} \approx 0.70068$$
Step7: Calculate the inverse cosine to find $m\angle S$
$$m\angle S = \arccos(0.70068) \approx 45.52^\circ$$
Step8: Round to the nearest tenth
$m\angle S \approx 45.5^\circ$
Answer:
$m\angle S = 45.5$