Type a trig equation, an exact value or an identity. Equations like 2sin²x − sin x − 1 = 0 get the general solution and the answers in [0, 2π) or 0° to 360°; values like cos 300° come from the unit circle and reference angles; identities such as tan x + cot x = sec x csc x are proved line by line.
How a trig equation is solved
The calculator chooses the method a teacher would, depending on what the equation contains:
| The equation has… | Example | Method |
|---|---|---|
| One trig function, to the first power | 2cos x = √2 | Isolate it, then use the reference angle and the quadrant signs |
| One function, squared | 2sin²x − sin x − 1 = 0 | Solve as a quadratic in sin x (factoring or the formula) |
| sin² with cos, or cos² with sin | 2cos²x + 3sin x − 3 = 0 | Pythagorean identity \sin^2 x + \cos^2 x = 1 turns it into one function |
| sin and cos to the same power | sin x = cos x | Divide by \cos x to get tan x |
| A common factor | sin 2x = cos x | Double-angle identity, factor, set each factor to 0 |
a\sin x + b\cos x = c | √3 sin x + cos x = 1 | Write as one sine wave R\sin(x + \alpha) |
| The same function of two angles | sin 3x = sin x | Equal angles or supplementary angles, plus full turns |
| sec, csc or cot | sec x = 2 | Take the reciprocal: \cos x = \frac{1}{2} |
When x appears both inside and outside a trig function (like x + sin x = 1) there is no exact method, so the solutions are found numerically and the steps say so.
The general solution
Trig functions repeat, so an equation usually has infinitely many solutions. Once one turn is solved, every other solution is a whole number of periods away (n is any integer):
| Equation | General solution |
|---|---|
\sin x = k | x = \alpha + 2\pi n or x = \pi - \alpha + 2\pi n |
\cos x = k | x = \pm\alpha + 2\pi n |
\tan x = k | x = \alpha + \pi n |
For \sin 2x = k the period of 2x is still 2\pi, so after dividing by 2 the solutions repeat every \pi – twice as many fit in one turn.
Exact values of the special angles
| Angle | 0 | 30° = π/6 | 45° = π/4 | 60° = π/3 | 90° = π/2 |
|---|---|---|---|---|---|
| sin | 0 | \frac{1}{2} | \frac{\sqrt{2}}{2} | \frac{\sqrt{3}}{2} | 1 |
| cos | 1 | \frac{\sqrt{3}}{2} | \frac{\sqrt{2}}{2} | \frac{1}{2} | 0 |
| tan | 0 | \frac{\sqrt{3}}{3} | 1 | \sqrt{3} | undefined |
For any other multiple of these angles, find the reference angle (the acute angle to the x-axis) and give the value the sign of its quadrant. The ASTC rule says which functions are positive: All in quadrant I, Sine in II, Tangent in III, Cosine in IV.
Example: \sin\frac{5\pi}{6}
\begin{aligned} \theta_{\text{ref}} &= \pi - \frac{5\pi}{6} = \frac{\pi}{6} && \text{quadrant II} \\ \sin\frac{5\pi}{6} &= +\sin\frac{\pi}{6} = \frac{1}{2} && \text{sine is positive in II} \end{aligned}Proving identities
Start with the more complicated side and change it, one identity at a time, until it matches the other side:
- Write tan, cot, sec and csc with sine and cosine.
- Combine fractions over a common denominator.
- Use
\sin^2 x + \cos^2 x = 1(or its forms1 - \cos^2 x = \sin^2 x,1 + \tan^2 x = \sec^2 x). - Factor and cancel, then name the result (
\frac{1}{\cos x} = \sec x).
If the two sides are different for some angle, the calculator shows that value as a counterexample.
Equations without trig functions go to the equation solver; complex numbers in polar form are on the complex number calculator.
SOHCAHTOA: trig in a right-angled triangle
- SOH: sin θ = opposite ÷ hypotenuse.
- CAH: cos θ = adjacent ÷ hypotenuse.
- TOA: tan θ = opposite ÷ adjacent.
Example: a ladder 5 m long leans at 60° to the ground. Its top reaches 5 × sin 60° = 5 × 0.866 = 4.33 m up the wall, and its foot is 5 × cos 60° = 2.5 m from the wall.
Degrees and radians
A full turn is 360° or 2π radians, so radians = degrees × π ÷ 180 and degrees = radians × 180 ÷ π. Common values: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π. One radian is about 57.3°. Type sin(30°) with the degree sign, or a plain number for radians.
Frequently asked questions
How do I get answers in degrees?
Add a degree interval such as “, 0 <= x < 360”, or write an angle with the ° key (sin(x + 30°) = 1/2). The answers are then given in degrees.
Why does sin x = 2 have no solution?
Sine and cosine are always between −1 and 1, so no angle has a sine of 2.
What is a reference angle?
The acute angle between the terminal side of the angle and the x-axis. Trig values of any angle equal ± the values of its reference angle; the sign depends on the quadrant.