high school Applications of Trigonometry

Applications of Trigonometry

In a triangle the sides and angles are linked, so a few known parts reveal the rest. The law of sines makes each side over the sine of its opposite angle equal the circumdiameter 2R, while the law of cosines c² = a² + b² − 2ab cos C is a generalized Pythagoras that reduces to it at a right angle. With the area formula S = (1/2)ab sin C you can find lengths and angles no ruler can reach, as in surveying or navigation. Adjust the vertex, sides, and included angle and the two laws fix a triangle.

Law of Sines — Circumscribed Circle

Position of vertex A50°

Law of Sines

asin A = bsin B = csin C = 2R

Each side over the sine of the opposite angle equals the circumdiameter

💡 Key Idea

①'Side / sine of opposite angle' = circumdiameter (2R)

②Knowing 2 angles + 1 side gives the rest

③Area: S = (1/2)ab sin C

④The on-canvas 2R number is pixels. Slider A is not the drawn angle

Law of Cosines — Generalized Pythagoras

Side a4

Side b3

Included angle C60°

Law of Cosines

c² = a² + b² - 2ab cos C

When C = 90° it reduces to the Pythagorean theorem

Area of a Triangle

Area (two sides + included angle)

S = 12 ab sin C

Two sides and the included angle

Heron's Formula

S = √(s(s-a)(s-b)(s-c)), s = a+b+c2

Compute area from three sides only

📐 Which Formula to Use?

①Two sides + included angle → S = (1/2)ab sin C

②Three sides only → Heron

③One side + two adjacent angles → law of sines first, then area

Real-life Applications

🌍 Where Trig Is Used

①Surveying: height of buildings/mountains

②Navigation: distance between two points

③Physics: force decomposition, vector composition

④Architecture: roof pitch, structural design

Wrap-up

Law of Sines

asin A = bsin B = csin C = 2R

side/sin = circumdiameter

Law of Cosines

c² = a² + b² - 2ab cos C

Generalization of Pythagoras

🎯 Exam Points

①Law of sines: angle-side correspondence (ASA, AAS)

②Law of cosines: SAS or SSS

③Area: S = (1/2)ab sin C

④C = 90° → cos C = 0 → Pythagoras

⑤Law of sines also gives circumradius R

Worked Examples & Past Exam

Example 1

In triangle ABC, a = 8 and sin A = 1/2. Find the circumradius R.

By the law of sines, a/sin A = 2R.

2R = asin A = 81/2

Compute to find R.

2R = 16, R = 8

▸ R = 8

Use the law of sines a/sin A = 2R to find the circumradius.

Example 2

With sides a = 5, b = 3 and included angle C = 60°, use the law of cosines to find c.

Substitute into the law of cosines c² = a² + b² − 2ab cos C.

c² = 5² + 3² - 2(5)(3) cos 60°

Substitute cos 60° = 1/2 and compute.

c² = 25 + 9 - 30 × 12 = 19, c = √19

▸ c = √19

Given two sides and the included angle, the law of cosines finds the third side.

exam-style

Find the area of a triangle with sides a = 4, b = 6 and included angle C = 30°.

①6

②12

③6√3

④3

⑤12√3

▸ ① 6

Substitute into the area formula S = (1/2)ab sin C.

S = 12 × 4 × 6 × sin 30°

Substitute sin 30° = 1/2.

S = 12 × 12 = 6

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Graphs of Trigonometric Functions

Trigonometric Equations

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