Law of sines [tex]\frac{\sin (A)}{a}=\frac{\sin (B)}{b}=\frac{\sin (C)}{c}[/tex]
How many distinct triangles can be formed for which [tex]m \angle A=75^{\circ}, a=2[/tex], and [tex]b=3[/tex] ?
One triangle can be formed where angle [tex]B[/tex] is about [tex]15^{\circ}[/tex].
No triangles can be formed.
Two triangles can be formed where angle [tex]B[/tex] is [tex]40^{\circ}[/tex] or [tex]140^{\circ}[/tex].
One triangle can be formed where angle about [tex]40^{\circ}[/tex].
Answer (1)
Use the Law of Sines to relate the given angle and sides: a s i n ( A ) = b s i n ( B ) .
Substitute the given values A = 7 5 ∘ , a = 2 , and b = 3 into the Law of Sines.
Calculate sin ( B ) = 2 3 s i n ( 7 5 ∘ ) ≈ 1.4489 .
Since 1"> sin ( B ) > 1 , no triangle can be formed: No triangles can be formed.
Explanation
Problem Setup and Law of Sines 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 . The Law of Sines states that a s i n ( A ) = b s i n ( B ) .
Applying the Law of Sines Using the Law of Sines, we have 2 s i n ( 7 5 ∘ ) = 3 s i n ( B ) . Solving for sin ( B ) , we get sin ( B ) = 2 3 s i n ( 7 5 ∘ ) .
Calculating sin(B) We calculate the value of sin ( B ) : sin ( B ) = 2 3 s i n ( 7 5 ∘ ) ≈ 2 3 × 0.9659 ≈ 1.4489 .
Determining the Number of Triangles Since the value of sin ( B ) must be between -1 and 1, and we found that sin ( B ) ≈ 1.4489 , which is greater than 1, no triangle can be formed with the given conditions.
Final Answer Therefore, no triangle can be formed with the given conditions.
Examples
In surveying, determining the possible triangle formations is crucial when mapping terrains. If surveyors measure an angle and two sides, they need to know if a unique triangle (and thus a unique terrain map) can be formed. If sin ( B ) exceeds 1, it indicates that the side opposite angle B is too long to form a triangle, meaning no such terrain configuration is possible with those measurements.
