Sin Cos Tan Calculator

The three primary trigonometric functions — sine, cosine, and tangent — relate angles to ratios of sides in a right triangle:

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin\theta}{\cos\theta}$

On the unit circle (radius 1, centered at origin), for an angle $\theta$ measured from the positive $x$-axis:

• $\cos\theta$ = $x$-coordinate of the point
• $\sin\theta$ = $y$-coordinate of the point
• $\tan\theta$ = slope of the terminal ray

Key properties:

• $\sin$ and $\cos$ have range $[-1, 1]$ and period $2\pi$
• $\tan$ has range $(-\infty, \infty)$ and period $\pi$
• $\tan\theta$ is undefined when $\cos\theta = 0$ (at $\frac{\pi}{2} + n\pi$)

The reciprocal functions are:
$\csc\theta = \frac{1}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \cot\theta = \frac{1}{\tan\theta}$

These six functions form the foundation of trigonometry and appear throughout mathematics, physics, engineering, and signal processing.

Method 1: Unit Circle (Exact Values)

Memorize key angles and their coordinates on the unit circle:

Method 2: Reference Angles

For angles beyond the first quadrant:

1. Find the reference angle (acute angle to the $x$-axis)
2. Determine the sign from the quadrant (ASTC rule: All, Sin, Tan, Cos)

ASTC Rule — which functions are positive:

• Quadrant I (0° to 90°): All positive
• Quadrant II (90° to 180°): Sin positive
• Quadrant III (180° to 270°): Tan positive
• Quadrant IV (270° to 360°): Cos positive

Example: $\sin(150°)$ — Reference angle is $180° - 150° = 30°$. In Quadrant II, sine is positive: $\sin(150°) = +\sin(30°) = \frac{1}{2}$.

Method 3: Sum and Difference Formulas

For non-standard angles, decompose into known angles:

$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$
$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

Example: $\cos(75°) = \cos(45° + 30°) = \cos 45° \cos 30° - \sin 45° \sin 30° = \frac{\sqrt{6} - \sqrt{2}}{4}$

Method 4: Graphing Transformations

For $y = A\sin(Bx + C) + D$:

• $|A|$ = amplitude
• $\frac{2\pi}{|B|}$ = period
• $-\frac{C}{B}$ = phase shift
• $D$ = vertical shift

Comparison: When to Use Each Method

• Wrong quadrant sign: $\cos(120°) = -\frac{1}{2}$, not $+\frac{1}{2}$. Always check which quadrant determines the sign.
• Confusing degrees and radians: $\sin(\pi) = 0$ (radians), but $\sin(180) \approx -0.80$ if interpreted as 180 radians. Be consistent with units.
• Forgetting tan is undefined: $\tan(90°)$ and $\tan(270°)$ are undefined (vertical asymptotes), not zero or infinity.
• Misapplying the sum formula: $\sin(A + B) \neq \sin A + \sin B$. You must use the correct expansion.
• Reference angle errors: The reference angle is always measured to the $x$-axis (not the $y$-axis), and is always positive and acute.