A graph-and-unit-circle walkthrough of the big three identity families students use constantly: reciprocal identities, Pythagorean identities, and sum/difference identities.
These say secant, cosecant, and cotangent are just reciprocals of cosine, sine, and tangent. They are especially useful when rewriting an expression into only sin and cos.
If cosine is the x-coordinate, secant is the reciprocal of that x-coordinate.
If sine is the y-coordinate, cosecant is the reciprocal of that y-coordinate.
Cotangent is the reciprocal of tangent: cot θ = 1 / tan θ.
Simplify: sin θ · csc θ
Rewrite using only sin and cos: tan θ · sec θ
The unit circle is built from x² + y² = 1. Since x = cos θ and y = sin θ, the main identity is sin²θ + cos²θ = 1. The other two come from dividing by cos²θ or sin²θ.
Tip: if an expression has sec² or csc², look for a way to swap it for 1 + tan² or 1 + cot².
Simplify: sec²θ − tan²θ
If sin θ = 3/5 and θ is in Quadrant I, find cos θ.
These let you find exact trig values for angles like 75°, 15°, and 105° by building them from special angles. Use the sliders to see how α and β combine.
For α = 45° and β = 30°: sin(α + β) ≈ 0.966, cos(α + β) ≈ 0.259.
Find exact value: sin 75°
Find exact value: cos 15°