Integrated Math 3 / Unit 6: Trigonometric Identities and Laws
6.6 Applications of the Laws
Notes
First draw and label a triangle. Choose the law by what is known.
- Side-angle pair known (AAS, ASA, SSA): Law of Sines.
- SAS or SSS: Law of Cosines.
- Bearings and distances, surveying across rivers, and land area are common settings.
Practice
Two ships leave a port. One sails 20 km and the other sails 30 km, and the angle between their paths is $60^\circ$. About how far apart are they?
$d^2 = 400 + 900 - 2(20)(30)\cos 60^\circ = 1300 - 600 = 700$, so $d \approx 26.5$.
Two surveyors stand 100 m apart on one bank of a river. The angles from the baseline to a tree on the other bank are $50^\circ$ at the first surveyor and $70^\circ$ at the second. About how far is the tree from the first surveyor?
The angle at the tree is $60^\circ$. The distance from the first surveyor is opposite the $70^\circ$ angle: $\displaystyle \frac{100\sin 70^\circ}{\sin 60^\circ} \approx 108.5$.
A triangular lot has two sides of 50 m and 70 m with a $40^\circ$ angle between them. What is its area to the nearest square meter?
$\displaystyle \frac{1}{2}(50)(70)\sin 40^\circ = 1750(0.6428) \approx 1125$.
Ready for more? Take the Unit 6 test.