Unformatted text preview: from C to the island. Using Theorem 10.4, we get sin (45◦ ) = y . After some rearranging,
√ we ﬁnd y = d sin (45◦ ) ≈ 9.66 22 ≈ 6.83 miles. Hence, the island is approximately 6.83 miles
from the coast. To ﬁnd the distance from Q to C , we note that β = 180◦ − 90◦ − 45◦ = 45◦ so
by symmetry,11 we get x = y ≈ 6.83 miles. Hence, the point on the shore closest to the island is
approximately 6.83 miles down the coast from the second observation point.
Sasquatch Island Sasquatch Island β β
d ≈ 9.66 miles 30◦ γ 45◦ Q P d ≈ 9.66 miles y miles 45◦ Shoreline
Q 5 miles C
x miles We close this section with a new formula to compute the area enclosed by a triangle. Its proof uses
the same cases and diagrams as the proof of the Law of Sines and is left as an exercise.
Theorem 11.4. Suppose (α, a), (β, b) and (γ, c) are the angle-side opposite pairs of a triangle.
Then the area A enclosed by the triangle is given by
A = bc sin(α) = ac sin(β ) = ab sin(γ )
Example 11.2.4. Find the area of the triangle in Example 11.2.2 number 1.
Solution. From our work in Example 11.2.2 number...
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