Sine Cosine Tangent Calculator

Trigonometry relates the angles of a triangle to the lengths of its sides. For any right-angled triangle, the primary functions are defined using the sides relative to a given acute angle: the opposite side, the adjacent side, and the hypotenuse. The sine function is the ratio of the opposite side to the hypotenuse (sin = opposite / hypotenuse). The cosine function compares the adjacent side to the hypotenuse (cos = adjacent / hypotenuse). Finally, the tangent function measures the ratio of the opposite side to the adjacent side (tan = opposite / adjacent), which is mathematically equivalent to dividing the sine by the cosine. When working with arbitrary angles, the calculator normalizes degrees into radians and applies cyclic adjustments to handle periodicity and vertical asymptotes for secondary functions like cotangent, secant, and cosecant.

Let us calculate the trigonometric values for a classic right triangle where the side adjacent to angle alpha is 3 units, and the side opposite to it is 4 units (the famous 3-4-5 right triangle). First, determine the hypotenuse using the Pythagorean theorem: c = sqrt(a^2 + b^2) = sqrt(4^2 + 3^2) = sqrt(16 + 9) = sqrt(25) = 5. Next, compute the sine of angle alpha by dividing the opposite side by the hypotenuse: sin(alpha) = 4 / 5 = 0.8. Then, calculate the cosine by dividing the adjacent side by the hypotenuse: cos(alpha) = 3 / 5 = 0.6. Finally, find the tangent by dividing the opposite side by the adjacent side: tan(alpha) = 4 / 3 = 1.3333. This combination completely defines the geometric proportions of the triangle.

Always verify whether your calculator input expects degrees or radians before computing values, as mixing these units will yield incorrect trigonometric outputs. Remember that the sine and cosine of any real angle can never exceed 1 or fall below -1, whereas the tangent can range infinitely from negative to positive infinity. When dealing with tangent and secant calculations near odd multiples of 90 degrees, keep watch for undefined values caused by division by zero.