Real-World Applications of Trigonometry
Apply the Law of Sines, Law of Cosines, and angles of elevation and depression to solve real-world problems involving non-right triangles and navigation.
Not every real-world triangle has a right angle. A surveyor measuring a triangular property, an air traffic controller tracking a plane's bearing, or an engineer designing a truss with no right angles — all need tools that go beyond SOHCAHTOA. The Law of Sines and Law of Cosines solve any triangle, and angles of elevation/depression are the trig of everyday observation. This lesson ties together everything from the course into practical problem-solving.
- Solve elevation and depression angle problems using SOHCAHTOA
- Apply the Law of Sines to find sides or angles in non-right triangles
- Apply the Law of Cosines when two sides and the included angle are known (SAS)
- Solve navigation problems using bearing and distance
Angle of Elevation/Depression: Draw a horizontal reference line, identify the angle, and apply SOHCAHTOA or tan to the resulting right triangle.
Law of Sines:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]Law of Cosines (generalization of Pythagorean Theorem):
\[ c^2 = a^2 + b^2 - 2ab\cos C \qquad (\text{use this form to find the third side}) \] \[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \qquad (\text{rearrange to find angle C}) \]When to Use Which Law
| Given | Use |
|---|---|
| Right triangle + angle | SOHCAHTOA |
| Right triangle + two sides | Pythagorean Theorem or inverse trig |
| AAS or ASA (two angles + one side) | Law of Sines |
| SSA (two sides + non-included angle) | Law of Sines (ambiguous case) |
| SAS (two sides + included angle) | Law of Cosines |
| SSS (all three sides) | Law of Cosines (to find angles) |
From a point 60 feet from the base of a building, the angle of elevation to the top is 55°. How tall is the building?
- Horizontal distance = adjacent = 60 ft; height = opposite; angle = 55°.
- \(\tan 55° = \frac{\text{height}}{60}\)
- \(\text{height} = 60 \times \tan 55° = 60 \times 1.428 \approx \mathbf{85.7 \text{ ft}}\)
In triangle ABC, angle A = 35°, angle B = 80°, and side a = 12. Find side b.
- Angle C = 180° − 35° − 80° = 65°
- \(\frac{b}{\sin B} = \frac{a}{\sin A}\) → \(\frac{b}{\sin 80°} = \frac{12}{\sin 35°}\)
- \(b = \frac{12 \times \sin 80°}{\sin 35°} = \frac{12 \times 0.985}{0.574} \approx \mathbf{20.6}\)
Two ships leave a port. Ship A travels 40 km on bearing 040°; Ship B travels 55 km on bearing 130°. The angle between their paths is 90°. How far apart are the ships?
\[ d^2 = 40^2 + 55^2 - 2(40)(55)\cos 90° = 1600 + 3025 - 0 = 4625 \] \[ d = \sqrt{4625} \approx \mathbf{68.0 \text{ km}} \]Since cos 90° = 0, the Law of Cosines reduces to the Pythagorean Theorem when the included angle is exactly 90° — confirming the connection.
- From a cliff 80 m high, the angle of depression to a boat is 22°. How far is the boat from the base of the cliff?
- In a triangle, sides a = 7, b = 10, and angle A = 40°. Use the Law of Sines to find angle B.
- In a triangle, sides a = 8, b = 11, and included angle C = 60°. Find side c using the Law of Cosines.
▶ Show Answers
- \(\tan 22° = 80/d\) → \(d = 80/\tan 22° \approx 80/0.404 \approx \mathbf{198 \text{ m}}\).
- \(\sin B = 10\sin40°/7 = 10(0.643)/7 \approx 0.918\) → \(B \approx \sin^{-1}(0.918) \approx \mathbf{66.8°}\).
- \(c^2 = 64 + 121 - 2(8)(11)\cos60° = 185 - 176(0.5) = 185 - 88 = 97\) → \(c \approx \mathbf{9.85}\).
- Using Law of Sines when Law of Cosines is needed: Law of Sines requires at least one matching side-angle pair. If you have SAS (two sides and the angle between them), you must use the Law of Cosines — Law of Sines won't work directly.
- Confusing angle of elevation and depression: Both are measured from the horizontal. Elevation looks up; depression looks down. The angle is always between the line of sight and a horizontal (not vertical) reference.
- Forgetting to subtract angles from 180° when solving triangles: Once you find one angle, always check that all three angles sum to 180° before finding the third side.
- Elevation/Depression: Draw horizontal reference, apply tan (or sin/cos) to the right triangle formed.
- Law of Sines: \(a/\sin A = b/\sin B = c/\sin C\) — use for AAS, ASA, SSA.
- Law of Cosines: \(c^2 = a^2 + b^2 - 2ab\cos C\) — use for SAS, SSS.
- When C = 90°, the Law of Cosines reduces to the Pythagorean Theorem.
Land surveyors use the Law of Sines and Cosines to calculate property boundaries, triangulate GPS reference points, and measure land area without direct access. Pilots and navigators use bearing-based trig to calculate cross-track error and course corrections. Drone operators and satellite positioning systems use elevation angles to compute altitude from ground measurements. Every field that involves triangulation — from archaeology to air traffic control — is built on these two laws.
Calculator Connection
The Elevation & Depression Angle Calculator solves line-of-sight problems instantly. The Law of Sines Calculator and Law of Cosines Calculator solve any non-right triangle with step-by-step work. The Bearing & Navigation Calculator handles compass-based direction problems.
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Real-World Applications of Trigonometry — Quiz
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