CalcElfCalcElf › Sides 7, 8, 9: angle opposite 7
Sides 7, 8, 9: angle opposite 7
The three side lengths of a triangle are 7, 8, and 9. Find the size of the angle opposite to side 7.
Answer
A = 48.189685°, area S = 26.832816
Step-by-step
- Given three sides a = 7, b = 8, c = 9With all three sides known, use the Law of Cosines to find an angle.
- cosA = (b² + c² − a²)/(2bc) = 0.66666667The Law of Cosines rearranged to solve for an angle from three sides.
- A = arccos(0.66666667) = 48.189685°A triangle's interior angle lies between 0 and 180 degrees, so the principal value of arccosine is enough
- S = ½·b·c·sinA = 26.832816After finding the angle, plug it into the two-sides-and-included-angle formula to get the area.
- Check: A + B + C = 180°The sum of interior angles of a triangle must be 180°, which can serve as an internal check for angle calculations.
Why it works
When all three sides are known and an angle is to be found, the corollary of the Law of Cosines should be used because cosine is monotonic between zero and one hundred eighty degrees, so a unique angle can be determined from its cosine value without producing two solutions. This is more reliable than the Law of Sines: in obtuse cases, the Law of Sines yields two candidate angles and requires additional judgment. After calculating the angle, substitute it back once using the sum of interior angles and once using the area formula to verify both steps simultaneously.
Common mistakes
- Using the Law of Sines to find an angle without checking for obtuse cases, leading to selecting the wrong angle.
- Misplacing the signs of the three terms in the numerator of the corollary of the Law of Cosines by writing plus instead of minus.
- Treating the result of an arccosine calculation as radians and writing only the numerical value while omitting the degree unit.