Complete trigonometric solution sets

1. Find the reference value.

   $$x_1=30^\circ$$

2. Reflect within the upper semicircle.

   $$x_2=180^\circ-30^\circ=150^\circ$$

30°,150°.

Checks and common pitfalls: Both values meet the interval; no extra full turn fits.

1. Obtain admissible sine values.

   $$\sin x=1\text{ or }-1/2$$

2. Recover all angle branches and deduplicate.

   $$x=90^\circ,210^\circ,330^\circ$$

90°,210°,330°.

Checks and common pitfalls: The sine value 1 has a single point in this cycle.

1. Preserve the zero factor branch.

   $$\cos x(2\sin x-1)=0$$

2. List both branch sets.

   $$x=\pi/2,3\pi/2;\quad x=\pi/6,5\pi/6$$

π/6, π/2, 5π/6, 3π/2; division loses the two cosine zeros.

Checks and common pitfalls: Checking only the divided equation cannot recover a discarded branch.

1. Identify the first zero.

   $$x=90^\circ$$

2. Include the opposite ray.

   $$x=270^\circ$$

90°,270°.

Checks and common pitfalls: The cosine graph crosses zero twice in a full cycle.

1. Find one valid reference solution.

   $$x=45^\circ$$

2. Add one tangent period within the interval.

   $$x=225^\circ$$

45°,225°.

Checks and common pitfalls: Using a 360° period alone would omit a valid tangent solution.

1. Apply the unit-circle bound.

   $$-1\le\sin x\le1$$

2. The requested value lies outside it.

   $$2>1$$

No real solutions.

Checks and common pitfalls: A range check prevents meaningless inverse-function input.

1. List admissible multiples.

   $$x=-\pi,0,\pi$$

2. Count distinct real inputs.

   $$N=3$$

3.

Checks and common pitfalls: Coterminal endpoint angles are different real inputs and both belong to this closed interval.

1. List solutions for the transformed angle.

   $$2x=30^\circ,150^\circ,390^\circ,510^\circ$$

2. Return to the original input.

   $$x=15^\circ,75^\circ,195^\circ,255^\circ$$

15°,75°,195°,255°.

Checks and common pitfalls: Using only one transformed cycle halves the complete solution set.

1. Restore the input by adding π/3.

   $$x=\pi,5\pi/3$$

2. Other periodic copies lie outside the interval.

   $$0\le x<2\pi$$

π and 5π/3.

Checks and common pitfalls: Shift first, then check the original domain rather than an assumed one-cycle phase interval.

1. Exclude a zero divisor by the equation itself.

   $$\cos x=0\Rightarrow\sin x=\pm1\ne0$$

2. Divide safely and use tangent period.

   $$\tan x=1\Rightarrow x=\pi/4,5\pi/4$$

π/4 and 5π/4; division is safe after the zero-case check.

Checks and common pitfalls: This contrasts with equations where the zero divisor actually supplies valid solutions.

1. The squared equation allows both signs.

   $$x=30^\circ,150^\circ,210^\circ,330^\circ$$

2. Reject the positive-sine candidates.

   $$x=210^\circ,330^\circ\text{ remain}$$

Reject 30° and 150°; retain 210° and 330°.

Checks and common pitfalls: Substitution checks validity after an implication that was not reversible.

1. Classify impossible and extremal levels.

   $$|k|>1:0;\quad |k|=1:1$$

2. Treat ordinary levels and the duplicated endpoint height.

   $$0<|k|<1:2;\quad k=0:3$$

Counts: 0,1,2,3 under the stated four cases.

Checks and common pitfalls: The endpoints are coterminal but distinct real inputs, so both count at zero.

1. Express the strict phase interval.

   $$\pi/6<\pi t/4<5\pi/6$$

2. Convert back to seconds and exclude equality times.

   $$2/3<t<10/3$$

Strictly above 4 m for t∈(2/3,10/3) s.

Checks and common pitfalls: At both endpoints the point is exactly 4 m high, so neither endpoint is included.