Revise: Trigonometric Functions and Their Graphs

y=sinx: range [−1,1], period 2π, passes through origin.

sin(π/6)=sin(5π/6)=0.5.

y=cosx: same shape as sine, starts at max (1) at x=0.

cos(0)=1, cos(π)=−1.

Asymptotes at π/2+kπ; period π.

tan(π/4)=1.

csc=1/sin, sec=1/cos, cot=1/tan.

sec(0)=1; csc(0) undefined.

y=A sin(B(x−C))+D: amplitude|A|, period 2π/B.

3sin(2(x−π/4)) → amp 3, period π.

Negative sign reflects; +D shifts vertically.

−sin(x)+4 has max 5.

arcsin(x): range [−π/2,π/2].

arcsin(0.5)=π/6.

arccos range [0,π]; arctan range (−π/2,π/2).

arccos(0.5)=60°; arctan(1)=45°.

d/dx[sinx]=cosx; d/dx[cosx]=−sinx.

f=sinx+cosx → f'(0)=1.

d/dx[tanx]=sec²x.

tan'(0)=1.

sin(π−θ)=sin(θ), a reflection symmetry.

sin(π/2)=1.

The (1,0) point on the unit circle at x=0 gives cosine's max.

cos(π/3)=0.5.

tan(x)=sin(x)/cos(x); undefined when cos(x)=0.

tan(0)=0.

Each reciprocal function is undefined where its partner is zero.

cot(π/4)=1.

Larger B compresses the graph (shorter period).

5sin(x) has amplitude 5.

Reflect first, then shift, matching the written order.

−cos(x)+2 → range [1,3].

The restricted range makes arcsin a genuine function.

arcsin(1)=90°.

Each inverse function has its OWN restricted range.

arccos(0)=90°.

Near x=0, sin(x) is nearly indistinguishable from y=x.

f=3sinx−2cosx → f'=3cosx+2sinx.

All four derived from sin/cos derivatives via the Quotient Rule.

sec'(0)=0.