Trigonometry: Right Triangles and the Unit Circle

Right Triangle Trigonometry

For a right triangle with angle θ, opposite side O, adjacent side A, hypotenuse H:

SOHCAHTOA:

• sin θ = O/H
• cos θ = A/H
• tan θ = O/A

Reciprocals:

• csc θ = 1/sin θ = H/O
• sec θ = 1/cos θ = H/A
• cot θ = 1/tan θ = A/O

Special Angle Values

| θ | sin θ | cos θ | tan θ | |---|-------|-------|-------| | 0° | 0 | 1 | 0 | | 30° | 1/2 | √3/2 | 1/√3 | | 45° | √2/2 | √2/2 | 1 | | 60° | √3/2 | 1/2 | √3 | | 90° | 1 | 0 | undefined |

The Unit Circle

A circle of radius 1 centered at the origin. Any point P on the circle:

• P = (cos θ, sin θ)
• θ measured counterclockwise from positive x-axis

Key Identities

Pythagorean Identities

• sin²θ + cos²θ = 1 ← Most important!
• 1 + tan²θ = sec²θ
• 1 + cot²θ = csc²θ

Angle Addition

• sin(A+B) = sinA cosB + cosA sinB
• cos(A+B) = cosA cosB - sinA sinB

Double Angle

• sin(2θ) = 2 sinθ cosθ
• cos(2θ) = cos²θ - sin²θ = 1 - 2sin²θ

Law of Sines & Cosines

For any triangle with sides a, b, c and opposite angles A, B, C:

Law of Sines: a/sin A = b/sin B = c/sin C

Law of Cosines: c² = a² + b² - 2ab cos C

(Law of cosines reduces to Pythagorean theorem when C = 90°)

Engineering Applications

• Resolving force vectors into components
• Finding angles in trusses
• Statics and dynamics problems
• Surveying and slope calculations

Summary

Trigonometry is the bridge between geometry and analysis. SOHCAHTOA, special angles, and sin²+cos²=1 appear daily in engineering. Know these cold.