Right Triangles & Trig Functions — Quick Notes

Right Triangle Basics

  • A right triangle has one angle of 90°, represented by a small box.
  • Sides: two legs (the non-hypotenuse sides) and the hypotenuse (the longest side).
  • Opposite and adjacent are defined with respect to a chosen angle (\theta); they depend on orientation.
  • Pythagorean theorem: a2+b2=c2a^2 + b^2 = c^2
  • Example: legs = 4 and 7; hypotenuse c=42+72=65c = \sqrt{4^2 + 7^2} = \sqrt{65}

Trigonometric Functions and Definitions

  • Sine: sin⁡θ=oppositehypotenuse\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}
  • Cosine: cos⁡θ=adjacenthypotenuse\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}
  • Tangent: tan⁡θ=oppositeadjacent\tan\theta = \frac{\text{opposite}}{\text{adjacent}}
  • Reciprocal functions:
    • Cosecant: csc⁡θ=1sin⁡θ=hypotenuseopposite\csc\theta = \frac{1}{\sin\theta} = \frac{\text{hypotenuse}}{\text{opposite}}
    • Secant: sec⁡θ=1cos⁡θ=hypotenuseadjacent\sec\theta = \frac{1}{\cos\theta} = \frac{\text{hypotenuse}}{\text{adjacent}}
    • Cotangent: cot⁡θ=1tan⁡θ=adjacentopposite\cot\theta = \frac{1}{\tan\theta} = \frac{\text{adjacent}}{\text{opposite}}
  • Key idea: knowing sin, cos, tan lets you deduce the other three via reciprocals; three functions determine all six.

Example: Triangle with legs 4 and 7

  • Opposite = 7, Adjacent = 4, Hypotenuse = c=42+72=65c = \sqrt{4^2 + 7^2} = \sqrt{65}
  • Sine: sin⁡θ=765=76565\sin\theta = \frac{7}{\sqrt{65}} = \frac{7\sqrt{65}}{65}
  • Cosine: cos⁡θ=465=46565\cos\theta = \frac{4}{\sqrt{65}} = \frac{4\sqrt{65}}{65}
  • Tangent: tan⁡θ=74\tan\theta = \frac{7}{4}
  • Cosecant: csc⁡θ=657\csc\theta = \frac{\sqrt{65}}{7}
  • Secant: sec⁡θ=654\sec\theta = \frac{\sqrt{65}}{4}
  • Cotangent: cot⁡θ=47\cot\theta = \frac{4}{7}

Special Angles: 45°, 30°, 60°

  • 45-45-90 triangle: two legs equal; if legs = 1, hypotenuse = 2\sqrt{2}.
  • For 45°:
    • sin⁡45∘=cos⁡45∘=22\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}
    • tan⁡45∘=1\tan 45^\circ = 1
    • csc⁡45∘=sec⁡45∘=2\csc 45^\circ = \sec 45^\circ = \sqrt{2}
    • cot⁡45∘=1\cot 45^\circ = 1
  • 30-60-90 triangle: sides in ratio 1:3:2 1 : \sqrt{3} : 2\, (opposite 30°, opposite 60°, hypotenuse)
  • For 30°:
    • sin⁡30∘=12\sin 30^\circ = \frac{1}{2}
    • cos⁡30∘=32\cos 30^\circ = \frac{\sqrt{3}}{2}
    • tan⁡30∘=13=33\tan 30^\circ = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
  • For 60°:
    • sin⁡60∘=32\sin 60^\circ = \frac{\sqrt{3}}{2}
    • cos⁡60∘=12\cos 60^\circ = \frac{1}{2}
    • tan⁡60∘=3\tan 60^\circ = \sqrt{3}
  • Reciprocals for 30° and 60°:
    • csc⁡30∘=2, sec⁡30∘=23=233, cot⁡30∘=3\csc 30^\circ = 2,\, \sec 30^\circ = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3},\, \cot 30^\circ = \sqrt{3}
    • csc⁡60∘=23=233, sec⁡60∘=2, cot⁡60∘=13=33\csc 60^\circ = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3},\, \sec 60^\circ = 2,\, \cot 60^\circ = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}

Unit Circle and Standard Position (brief)

  • Unit circle: radius = 1.
  • Point on circle at angle θ: $(\cos\theta, \sin\theta)$.
  • Standard position: vertex at origin, initial side along +x axis; rotating to the terminal side defines θ.
  • Relations on unit circle:
    • cos⁡θ=x,sin⁡θ=y\cos\theta = x,\quad \sin\theta = y where the point is $(x,y)$ on the circle.
    • Reciprocal relations: csc⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=cos⁡θsin⁡θ\csc\theta = \frac{1}{\sin\theta},\quad \sec\theta = \frac{1}{\cos\theta},\quad \cot\theta = \frac{\cos\theta}{\sin\theta}
  • Note: Radiant values will be introduced later; current focus is degrees.

Quick Recap / Key Takeaways

  • In any right triangle, define trig functions relative to the chosen angle (\theta) using the three basic ratios.
  • The other three trig functions are reciprocals of sine, cosine, and tangent.
  • Special triangles give standard values (45-45-90 and 30-60-90).
  • Unit circle connects trig values to coordinates: (\cos\theta = x), (\sin\theta = y) for a point on the circle of radius 1.