Trigonometry · eight-question check

1. Normalisation divides by positive r.

   $$r>0$$

2. Both signs remain negative.

   $$(\sin\theta,\cos\theta)=(-,-)$$

D: (−,−).

Checks and common pitfalls: Sine and cosine are signed coordinates, not positive lengths.

1. Match angle units.

   $$\pi/6=30^\circ$$

2. Read the vertical coordinate.

   $$\sin(\pi/6)=1/2$$

A: 1/2.

Checks and common pitfalls: The value is not the angle measure itself.

1. The combined angle is 60°.

   $$30^\circ+30^\circ=60^\circ$$

2. Read its unit-circle height.

   $$\sin60^\circ=\sqrt3/2$$

B: √3/2.

Checks and common pitfalls: Adding the two sine values incorrectly gives 1.

1. Substitute into the unaltered quotient.

   $$(1-1)/0=0/0$$

2. Simplification cannot restore this input.

   $$x=0\text{ excluded}$$

C: undefined.

Checks and common pitfalls: The simplified sine formula has a larger domain.

1. Factor the full angle.

   $$2(x+\pi/6)$$

2. Read the leftward displacement.

   $$h=-\pi/6$$

C: left π/6.

Checks and common pitfalls: Both division by frequency and direction matter.

1. Compare shifted outputs.

   $$f(x+T)=4=f(x)$$

2. There is no smallest positive choice.

   $$T/2<T\quad(T>0)$$

D: none exists.

Checks and common pitfalls: Zero is not a positive period, and choosing 2π is not least.

1. Find the reference angle.

   $$60^\circ$$

2. Reflect across the x-axis.

   $$360^\circ-60^\circ=300^\circ$$

B: {60°,300°}.

Checks and common pitfalls: The inverse principal value alone omits the reflected solution.

1. Transform the lower and upper bounds.

   $$0\le2x<720^\circ$$

2. The upper endpoint remains excluded.

   $$u\in[0,720^\circ)$$

C: [0,720°).

Checks and common pitfalls: Keeping [0,360°) would miss one phase cycle.