The fundamental identities
An identity is true for every value of the angle, unlike an equation which is only true for particular values. Three identities are essential and follow directly from the definitions and Pythagoras' theorem.
Key idea
sin2θ + cos2θ = 1, tan θ = sin θ / cos θ, and 1 + tan2θ = sec2θ.
From the first identity we get the useful rearrangements sin2θ = 1 − cos2θ and cos2θ = 1 − sin2θ, which let you replace one squared ratio by the other when simplifying or solving.
Solving trigonometric equations
To solve an equation over a given range: find the principal (basic) angle from the positive value, then use the CAST rule (or the graph) to list every solution in the interval. Remember that sine is positive in the first and second quadrants, cosine in the first and fourth, and tangent in the first and third.
Worked example
Solve sin θ = 0.5 for 0° ≤ θ ≤ 360°.
The basic angle is sin−1(0.5) = 30°. Sine is positive in the first and second quadrants, so θ = 30° and θ = 180° − 30° = 150°.
Remember
- Prove an identity by working on one side until it equals the other.
- Always state the range and give every solution within it.
- Replace 1 − cos2θ with sin2θ (and vice versa) to simplify.