$\Delta JKL$ has $j=7, k=11$, and $m \angle J=18^{\circ}$. Complete the statements to determine all possible measures of angle K.
Triangle JKL meets the $\square$ criteria, which means it is the ambiguous case.
Substitute the known values into the law of $\operatorname{sines}: \frac{\sin \left(18^{\circ}\right)}{7}=\frac{\sin (K)}{11}$.
Cross multiply: $11 \sin \left(18^{\circ}\right)=$ $\square$
Solve for the measure of angle K, and use a calculator to determine the value.
Round to the nearest degree: $m \angle K \approx$ $\square$ :
However, because this is the ambiguous case, the measure of angle K could also be $\square$ .
Answer (1)
Recognize the given information as the ambiguous SSA case.
Apply the Law of Sines to set up the equation: 7 s i n ( 1 8 ∘ ) = 11 s i n ( K ) .
Solve for sin ( K ) and find the first possible angle K ≈ 2 9 ∘ .
Determine the second possible angle K 2 = 18 0 ∘ − K 1 ≈ 15 1 ∘ , and verify its validity.
The possible measures of angle K are 2 9 ∘ , 15 1 ∘ .
Explanation
Problem Analysis We are given a triangle Δ J K L with side j = 7 , side k = 11 , and angle J = 1 8 ∘ . We want to find all possible measures of angle K . This is an SSA (side-side-angle) triangle, which is known as the ambiguous case because there might be zero, one, or two possible solutions for the angles and sides of the triangle.
Identifying the Ambiguous Case The triangle JKL meets the SSA criteria, which means it is the ambiguous case.
Applying the Law of Sines We use the Law of Sines to relate the sides and angles: j sin ( J ) = k sin ( K ) Substituting the given values, we have: 7 sin ( 1 8 ∘ ) = 11 sin ( K )
Cross Multiplication Cross-multiplying, we get: 11 sin ( 1 8 ∘ ) = 7 sin ( K ) So, the missing value is 7 sin ( K ) .
Solving for Angle K Now, we solve for sin ( K ) :
sin ( K ) = 7 11 sin ( 1 8 ∘ ) Using a calculator, we find the value of K :
K = arcsin ( 7 11 sin ( 1 8 ∘ ) ) K ≈ arcsin ( 7 11 × 0.3090 ) ≈ arcsin ( 0.4859 ) K ≈ 29.0 5 ∘ Rounding to the nearest degree, we get m ∠ K ≈ 2 9 ∘ .
Finding the Second Possible Angle Since this is the ambiguous case, there might be another possible solution for angle K . We find the second possible angle by subtracting the first angle from 18 0 ∘ :
K 2 = 18 0 ∘ − K 1 ≈ 18 0 ∘ − 2 9 ∘ = 15 1 ∘ Now, we check if this second angle is valid. The sum of angles in a triangle must be 18 0 ∘ . So, we check if J + K 2 < 18 0 ∘ :
1 8 ∘ + 15 1 ∘ = 16 9 ∘ < 18 0 ∘ Since the sum is less than 18 0 ∘ , the second angle is a valid solution.
Final Answer Therefore, the measure of angle K could also be approximately 15 1 ∘ .
Examples
The Law of Sines, especially in the ambiguous case, is useful in navigation and surveying. For example, if a surveyor knows the distance to a landmark from two different points and the angle at one of those points, they can use the Law of Sines to determine the possible locations of the landmark. This is crucial for creating accurate maps and determining property boundaries. The ambiguous case highlights that sometimes multiple solutions are possible, requiring further measurements to pinpoint the exact location.
