Laws of Sines and Cosines Quiz

Grade 10 · Math · Quiz

Test your ability to apply the Law of Sines and Law of Cosines to solve triangles.

Questions

1. In a triangle, side a = 8, angle A = 30°, angle B = 45°. Using the Law of Sines, side b =

   1. 8√2
   2. 4√2
   3. 8√2/√3
   4. 4
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   Answer: 8√2

   a/sin(A)=b/sin(B) → 8/sin(30°)=b/sin(45°) → 8/(0.5)=b/(√2/2) → 16=b/(√2/2) → b=16·(√2/2)=8√2.

2. You can use the Law of Sines to find a missing side when given SAS (two sides and their included angle).

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   Answer: False

   SAS requires Law of Cosines because you don't have a side-angle pair needed by Law of Sines.

3. The Law of Cosines: c² = a² + b² - 2ab · cos(___)

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   Answer: C

   The Law of Cosines c²=a²+b²-2ab·cos(C) relates side c to angle C, the angle opposite side c.

4. In a triangle with sides a=5, b=7, c=8, find cos(C) using the Law of Cosines:

   1. 1/7
   2. 10/70
   3. 1/14
   4. 5/7
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   Answer: 1/14

   cos(C)=(a²+b²-c²)/(2ab)=(25+49-64)/(2·5·7)=10/70=1/7.

5. In the ambiguous case (SSA), if the given side is shorter than the height from the opposite vertex, there are two possible triangles.

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   Answer: False

   If the given side is shorter than the height, there are ZERO triangles (not two). Two triangles occur when the side is between the height and the other given side.

6. What case name applies when you know all three sides of a triangle?

   1. AAS
   2. SSA
   3. SAS
   4. SSS
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   Answer: SSS

   SSS = three sides given. Use Law of Cosines to find each angle.

7. If angle A = 60°, a = 9, b = 9, then using the Law of Sines, sin(B) = ___.

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   Answer: sin(60°)

   a/sin(A)=b/sin(B) → 9/sin(60°)=9/sin(B) → sin(B)=sin(60°). Equal sides → equal opposite angles.

8. The sum of all angles in any triangle is always 180°.

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   Answer: True

   The angle sum property (A+B+C=180°) holds for all triangles in Euclidean geometry, regardless of type.

9. Which formula rearranges the Law of Cosines to find angle C when all three sides are known?

   1. cos(C) = (a+b-c)/(2ab)
   2. cos(C) = (a²+b²-c²)/(2ab)
   3. cos(C) = (a²-b²+c²)/(ab)
   4. cos(C) = c²/(a²+b²)
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   Answer: cos(C) = (a²+b²-c²)/(2ab)

   Rearrange c²=a²+b²-2ab·cos(C): 2ab·cos(C)=a²+b²-c² → cos(C)=(a²+b²-c²)/(2ab).

10. The Law of Sines works for all triangles, including obtuse triangles.

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    Answer: True

    The Law of Sines a/sinA=b/sinB=c/sinC is valid for all triangles — acute, right, and obtuse.

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