Inverse Trig Calculator

Q: What is arcsin of 1/2?

arcsin(1/2) = 30°, or π/6 radians. The equation sin x = 1/2 also has the solution 150° in one turn.

Q: Why does my calculator give only one answer?

An inverse function returns only the principal value. Other solutions come from the symmetry and period of the function.

Q: What is the range of arccos?

Arccos returns angles from 0° to 180°, or 0 to π radians, so arccos of a negative number is an obtuse angle.

Q: How do I find arcsec or arccsc on a calculator?

Use the reciprocal: arcsec k = arccos(1/k) and arccsc k = arcsin(1/k), for |k| ≥ 1.

Q: What does the n mean in the general solution?

n stands for any integer. Each choice of n gives another angle that solves the equation.

Q: Can I type √3/2 into the calculator?

Yes. Surds such as √3/2, sqrt(3)/2, 2√3/3 and 1/√2 are recognised, and standard values give exact angles.

Q: Why does sin x = 2 have no solution?

The sine of a real angle always lies between −1 and 1, so no angle has a sine of 2.

About the Inverse Trig Calculator

An inverse trigonometric function answers the question “which angle has this sine, cosine or tangent?” Calculators give a single answer, the principal value, but a trigonometric equation such as sin x = 1/2 has infinitely many solutions because the functions repeat. Exams usually ask for all solutions in an interval, or for the general solution.

This calculator gives all three. It recognises the standard exact values, so √2/2 returns exactly 45° or π/4 rather than a long decimal, and it accepts surds typed as √3/2, sqrt(3)/2 or 1/√2. Reciprocal functions are reduced to sine, cosine or tangent first, and every solution is substituted back as a check.

What are inverse trigonometric functions?

Sine, cosine and tangent are not one-to-one: many angles share the same sine. To define an inverse, each function is restricted to one interval where it takes every value exactly once. Arcsin returns an angle in [−90°, 90°], arccos in [0°, 180°] and arctan in (−90°, 90°). These ranges are the principal branches.

Arccsc, arcsec and arccot are defined the same way, using the reciprocal relationships. For example, arcsec 2 is the angle in [0°, 180°] whose cosine is 1/2, which is 60°. The usual convention, followed here, takes arccot in (0°, 180°), so arccot of a negative number is an obtuse angle.

How to use the Inverse Trig Calculator

Choose the function in the equation, such as sin or sec.
Type the value. Fractions, decimals and surds like −√3/2 or 2√3/3 are accepted.
Choose whether to show answers mainly in degrees or radians.
Press Solve.
Read the principal value, the solutions in one turn and the general solution.

Formulas

Principal ranges

\arcsin: [-\tfrac{\pi}{2}, \tfrac{\pi}{2}],\quad \arccos: [0, \pi],\quad \arctan: (-\tfrac{\pi}{2}, \tfrac{\pi}{2})

General solution of sin x = k

x = \arcsin k + 360^\circ n \ \text{or}\ x = 180^\circ - \arcsin k + 360^\circ n

General solution of cos x = k

x = \pm \arccos k + 360^\circ n

General solution of tan x = k

x = \arctan k + 180^\circ n

Reciprocal functions

\operatorname{arccsc} k = \arcsin \tfrac1k,\quad \operatorname{arcsec} k = \arccos \tfrac1k

Worked example: sin x = 1/2

The worked example solves $\sin x = \tfrac12$ . The principal value is $\arcsin \tfrac12 = 30° = \tfrac{\pi}{6}$ . Sine is also positive in the second quadrant, so $180° - 30° = 150°$ is a second solution in one turn.

The general solution is $x = 30° + 360°n$ or $x = 150° + 360°n$ , where n is any integer. In radians that is $x = \tfrac{\pi}{6} + 2\pi n$ or $x = \tfrac{5\pi}{6} + 2\pi n$ .

The solution below is generated by the same engine that powers the calculator above.

Final answer

\operatorname{arcsin}\left(1/2\right) = 30^\circ = \frac{\pi}{6},\quad x = 30^\circ + 360^\circ n \ \text{or}\ x = 150^\circ + 360^\circ n

How we read your problem

Find the angle whose sin is 1/2: the principal value of arcsin, every solution of sin(x) = 1/2 in one turn, and the general solution.

\sin x = 1/2

Step-by-step solution

Principal value of arcsin, taken in the range [−π/2, π/2]
30° is a special angle, so the answer is exact.
$\operatorname{arcsin}\left(1/2\right) = 30^\circ = \frac{\pi}{6}$
Sine is also positive or negative in a second quadrant, so use x and 180° − x
$\sin(180^\circ - x) = \sin x$
Solutions in one full turn, 0° ≤ x < 360°
$x = 30^\circ,\ 150^\circ \quad (\frac{\pi}{6},\ \frac{5\pi}{6})$
General solution, n any integer
$x = 30^\circ + 360^\circ n \ \text{or}\ x = 150^\circ + 360^\circ n$

Formulas used

Principal range of arcsin
$\operatorname{arcsin}: [-\tfrac{\pi}{2}, \tfrac{\pi}{2}]$

An inverse trigonometric function returns one angle, the principal value. Because the trigonometric functions repeat, the equation sin(x) = 1/2 has infinitely many solutions; the calculator lists those in one turn and the general pattern.

Verification

Substituting each solution back: sin(30°) ≈ 0.5; sin(150°) ≈ 0.5.

Detailed explanation

Using quadrants to find the second solution

Within one turn, sine and cosine each take most values twice. Sine is positive in quadrants I and II and negative in III and IV, which is why the partner of an angle θ is 180° − θ. Cosine is positive in quadrants I and IV, so the partner of θ is 360° − θ, the reflection across the x-axis.

For $\cos x = -\tfrac12$ , the principal value is 120°. Its reflection is 240°, so the solutions in one turn are 120° and 240°, and the general solution is $x = \pm 120° + 360°n$ .

Tangent and cotangent repeat every 180°

Tangent has period 180°, so once one solution is known, the others are found by adding or subtracting 180°. For $\tan x = -1$ , the principal value is −45°, giving 135° and 315° in one turn, and $x = -45° + 180°n$ in general.

Cotangent behaves the same way. The equation $\cot x = 0$ is special because it cannot be rewritten as $\tan x = 1/0$ ; instead, cotangent is zero exactly where cosine is zero, at 90° and 270°.

When there is no solution

Sine and cosine only produce values between −1 and 1, so $\sin x = 2$ has no real solution. Cosecant and secant are their reciprocals, so they never take values strictly between −1 and 1; $\sec x = 0.5$ is impossible for the same reason.

Tangent and cotangent take every real value, so an equation such as $\tan x = 1000$ always has solutions, here about 89.943° plus multiples of 180°.

Common mistakes

Stopping at the principal value when the question asks for all solutions in an interval.
Using 180° − θ for cosine instead of 360° − θ.
Adding 360° instead of 180° to tangent solutions, which misses half of them.
Reading sin⁻¹ x as 1/sin x; that is csc x, not arcsin x.
Leaving the calculator in the wrong angle mode when checking.

Tips

Sketch the unit circle and mark where the coordinate matches the value.
Learn the exact values for 30°, 45° and 60°; most textbook equations use them.
For an interval other than one turn, add or subtract the period until you leave the interval.
Rewrite csc, sec and cot equations with sin, cos and tan before solving.
Use the unit circle calculator to go the other way, from an angle to its trig values.

Frequently asked questions

What is arcsin of 1/2?