Definition
Circle with center O and radius 1. Any angle x corresponds to point M(cos x, sin x).
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Lesson overview
This lesson brings together the PDF, study notes and chapter flow so you can revise Trigonometric Calculus without losing the thread.
Definition
Circle with center O and radius 1. Any angle x corresponds to point M(cos x, sin x).
Fundamental relation
cos²(x) + sin²(x) = 1 for all x ∈ .
Periodicity
cos and sin have period 2π: cos(x + 2π) = cos(x), sin(x + 2π) = sin(x).
0, π/6, π/4, π/3, π/2
cos(0)=1 sin(0)=0 | cos(π/6)=√3/2 sin(π/6)=1/2 | cos(π/4)=√2/2 sin(π/4)=√2/2 | cos(π/3)=1/2 sin(π/3)=√3/2 | cos(π/2)=0 sin(π/2)=1
tan(x) = sin(x)/cos(x)
Defined for cos(x) ≠ 0, so x ≠ π/2 + kπ.
Addition formulas
cos(a+b) = cos a·cos b − sin a·sin b sin(a+b) = sin a·cos b + cos a·sin b
Double angle formulas
cos(2x) = cos²x − sin²x = 2cos²x − 1 = 1 − 2sin²x sin(2x) = 2·sin x·cos x
Linearization formulas
cos²x = (1 + cos 2x)/2 sin²x = (1 − cos 2x)/2
cos is even
cos(−x) = cos(x) for all x.
sin is odd
sin(−x) = −sin(x) for all x.
Complementary
cos(π/2 − x) = sin(x) and sin(π/2 − x) = cos(x).
Theorem 1 - Trigonometry 1
Application of theorem 1.
Theorem 2 - Trigonometry 1
Application of theorem 2.
Theorem 3 - Trigonometry 1
Application of theorem 3.
Theorem 4 - Trigonometry 1
Application of theorem 4.
Theorem 5 - Trigonometry 1
Application of theorem 5.
Problem-solving method
Identify data, set formulas, solve step by step, then verify sign, unit, and coherence.
Common mistakes
Forgetting domain constraints, mixing equality/equivalence, and losing solutions during transformations.