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Trigonometric equations

Reference angles and the infinitely many solutions.

sin x = ½ is true at π/6, at 5π/6, and again every full turn. Solve on one period from the unit circle, then add 2πn for every integer n. The graph shows why the solutions repeat.

Kushanda muenzaniso · sin(x) = 1/2

Trigonometry · Trigonometric-Equations

Solve sin(x) = 1/2

\sin{\left(x \right)} = \frac{1}{2}

Step by step

  1. \sin{\left(x \right)} = \frac{1}{2}

    Start from the equation as given.

  2. \sin{\left(x \right)} - \frac{1}{2} = 0

    Bring everything to one side.

  3. \sin{\left(x \right)} - \frac{1}{2} = 0

    Isolate the trig function, then read the reference angle off the unit circle. Solutions repeat every period.

  4. \left\{2 n \pi + \frac{\pi}{6}\; \middle|\; n \in \mathbb{Z}\right\} \cup \left\{2 n \pi + \frac{5 \pi}{6}\; \middle|\; n \in \mathbb{Z}\right\}

    General solution over the reals (n is any integer).

  5. x = \frac{\pi}{6} \approx 0.52360 ,\; x = \frac{5 \pi}{6} \approx 2.6180

    Solve for the variable.

Kuratidza mhinduro
x = \frac{\pi}{6} ,\; x = \frac{5 \pi}{6}

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Maitiro: Trigonometric equations

  1. Start from the equation as given.
  2. Bring everything to one side.
  3. Isolate the trig function, then read the reference angle off the unit circle. Solutions repeat every period.
  4. General solution over the reals (n is any integer).
  5. Solve for the variable.

Mabvunzo vanhu vanobvunza

Why radians instead of degrees?

A radian is the angle whose arc equals the radius, so it is a pure ratio rather than an arbitrary 1/360 of a turn. With radians the derivative of sin x is exactly cos x; with degrees a factor of π/180 appears everywhere.

Why does sin x = 1/2 have infinitely many solutions?

Sine repeats every full turn, and within one turn it reaches 1/2 twice (at π/6 and 5π/6). Add any whole number of turns to either and it is still true.

How do I remember the exact values?

Two triangles: the 45-45-90 with sides 1, 1, √2 and the 30-60-90 with sides 1, √3, 2. Every value for 30°, 45° and 60° is a ratio of those sides; symmetry gives the rest of the circle.

More in Trigonometry

The unit circleTrigonometric identitiesDegrees and radiansRight-triangle trigonometry (SOH-CAH-TOA)Law of sines and law of cosinesGraphs of sine, cosine and tangentInverse trigonometric functions