Question
9. Find side \( c \), if \( a=23, b=20 \), and \( \mathrm{m} \angle \mathrm{C}=47^{\circ} \). 27.503 17.366 39.452 16.005 All hanged saved . .
Ask by Mathis Lowe. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The length of side \( c \) is approximately 17.366.
Solution
To find side \( c \) in the triangle where \( a = 23 \), \( b = 20 \), and \( \angle C = 47^\circ \), we can use the **Law of Cosines**. The formula is:
\[
c^2 = a^2 + b^2 - 2ab \cos C
\]
Plugging in the given values:
\[
c^2 = 23^2 + 20^2 - 2 \times 23 \times 20 \times \cos(47^\circ)
\]
\[
c^2 = 529 + 400 - 920 \times \cos(47^\circ)
\]
\[
\cos(47^\circ) \approx 0.68199836
\]
\[
c^2 = 929 - 920 \times 0.682
\]
\[
c^2 = 929 - 627.44 = 301.56
\]
\[
c \approx \sqrt{301.56} \approx 17.366
\]
**Therefore, the length of side \( c \) is approximately 17.366.**
**Answer:** 17.366
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Simplify this solution Bonus Knowledge
To find side \( c \) using the Law of Cosines, we have the formula: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] Plugging in the values \( a = 23 \), \( b = 20 \), and \( C = 47^\circ \): \[ c^2 = 23^2 + 20^2 - 2 \cdot 23 \cdot 20 \cdot \cos(47^\circ) \]. Calculating this will give you the measure of side \( c \) approximately \( 27.503 \). In practical terms, the Law of Cosines is particularly handy in navigation and physics! If you're ever charting a course or inquiring about forces acting on an object at angles, this formula comes into play, making it essential for real-world applications involving triangles, whether in building designs or even triangular road networks!
