Math Problem Statement
Solution
To solve this triangle, we'll use the given information:
- Side inches,
- Side inches,
- Angle .
The unknowns we need to find are:
- The length of side ,
- The measure of angle ,
- The measure of angle .
Step-by-Step Solution
-
Finding side : We can apply the Law of Cosines: Substitute , , and .
-
Finding angle : Once is found, we can use the Law of Sines: Solve for using the calculated value of .
-
Finding angle : Using the fact that the sum of angles in a triangle is :
I'll calculate these values.The solutions are as follows:
- The length of side : inches (rounded to the nearest thousandth),
- The measure of angle : (rounded to the nearest hundredth),
- The measure of angle : (rounded to the nearest hundredth).
Would you like more detailed steps for any part of the solution, or have any questions?
Here are some related questions for further exploration:
- How would the solution change if angle were obtuse?
- What other trigonometric rules could solve this triangle?
- How would you solve it if only two sides and an angle were given?
- Can the Law of Sines give multiple solutions?
- What if side were shorter than side ?
Tip: Remember to check the range of trigonometric values when using inverse functions to avoid ambiguous cases in triangles.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Law of Cosines
Law of Sines
Formulas
Law of Cosines: b^2 = a^2 + c^2 - 2 * a * c * cos(B)
Law of Sines: sin(A) / a = sin(B) / b
Sum of Angles in a Triangle: C = 180° - B - A
Theorems
Law of Cosines
Law of Sines
Suitable Grade Level
Grades 10-12