How many distinct triangles can be formed for which [tex]$m \angle A=75^{\circ}, a=2$[/tex], and [tex]$b=3$[/tex]?
Answer (1)
Use the Law of Sines to relate the given angle A and sides a and b to find sin ( B ) .
Calculate sin ( B ) = 2 3 s i n ( 7 5 ∘ ) ≈ 1.449 .
Since 1"> sin ( B ) > 1 , no triangle can be formed.
Conclude that No triangles can be formed .
Explanation
Problem Analysis We are given angle A = 7 5 ∘ , side a = 2 , and side b = 3 . We want to determine how many distinct triangles can be formed with these conditions. We will use the Law of Sines to find the possible values for angle B .
Applying the Law of Sines Using the Law of Sines, we have: a sin ( A ) = b sin ( B ) Substituting the given values: 2 sin ( 7 5 ∘ ) = 3 sin ( B ) Solving for sin ( B ) :
sin ( B ) = 2 3 sin ( 7 5 ∘ )
Calculating sin(B) We found that sin ( B ) ≈ 1.449 . Since the value of sin ( B ) must be between -1 and 1, and we have 1"> sin ( B ) > 1 , no triangle can be formed with the given conditions.
Determining the Number of Triangles Since 1"> sin ( B ) > 1 , there are no possible values for angle B , and therefore no triangle can be formed.
Examples
In architecture, when designing triangular structures, it's crucial to verify if the given side lengths and angles can actually form a triangle. If the sine of an angle, calculated using the Law of Sines, exceeds 1, it indicates that the triangle cannot exist with the specified measurements. This ensures that the design is geometrically feasible before construction begins.
