Trigonometric Functions: Sine, Cosine, and Tangent

Trigonometric functions turn an angle into a number. On the unit circle, the point reached by a directed angle $\theta$ has horizontal coordinate $\cos \theta$ and vertical coordinate $\sin \theta$. The tangent is their quotient whenever the horizontal coordinate is not zero. This viewpoint joins right-triangle ratios, circular motion and periodic graphs in one model instead of treating them as unrelated formulas.

How an angle becomes three function values

Start on the positive horizontal axis and rotate counterclockwise for a positive angle or clockwise for a negative one. The unit-circle point determines cosine and sine directly. Tangent is the slope of the line through the origin and that point, so it is undefined at odd multiples of 90°, when the line is vertical. For an acute angle in a right triangle, the same coordinates reproduce adjacent over hypotenuse, opposite over hypotenuse, and opposite over adjacent.

The input may be written in degrees or radians. A full turn is 360° or $2\pi$ radians. Calculus and most higher mathematics use radians because they measure an angle by arc length divided by radius. Whatever unit you choose, set the calculator to that unit before evaluating a value.

Explore sine, cosine and tangent

Move the slider to follow the point around the unit circle and compare all three values.

Angle

Domains, ranges, and repeated graphs

Sine and cosine accept every real angle and always stay between negative one and one. Both repeat after $2\pi$. Sine is odd, which means changing the sign of the angle changes the sign of the output. Cosine is even, so it has the same value at opposite angles. Tangent accepts every real angle except $\frac{\pi }{2}+k\pi$, has every real number as a possible output, and repeats after $\pi$.

These facts explain the graphs. Sine crosses zero at integer multiples of $\pi$. Cosine reaches one at full turns and negative one after a half-turn. Tangent crosses zero at integer multiples of $\pi$ and approaches vertical asymptotes halfway between them. An asymptote is not a very large function value at the excluded angle; the function simply has no value there.

Exact values come from a reference angle and a quadrant

The familiar 30°, 45° and 60° values come from equilateral and isosceles right triangles. To evaluate a larger or negative angle, first find a coterminal angle between 0° and 360°. Then identify its reference angle, the acute angle to the horizontal axis. The reference angle gives the magnitude, while the quadrant gives the sign. Cosine is negative on the left half of the circle, sine is negative below the horizontal axis, and tangent is positive where sine and cosine share a sign.

Worked example: evaluate the three functions at 150°

The angle 150° lies in quadrant II and has reference angle 30°. At 30°, sine is one half and cosine is the square root of three over two. In quadrant II the vertical coordinate is positive and the horizontal coordinate is negative. Divide sine by cosine for tangent, then rationalize the denominator if an exact form is required.

The result also provides a quick check: the point must satisfy ${\cos }^2\theta +{\sin }^2\theta =1$, and a positive number divided by a negative number must give a negative tangent. For triangle solving with an opposite side-angle pair, continue with the law of sines.

What changes when the graph is transformed

In a rule such as $y=A\sin \left(B(x-C)\right)+D$, the absolute value of $A$ is the amplitude, $\frac{2\pi }{\left|B\right|}$ is the period, $C$ is the horizontal shift and $D$ is the midline. Cosine uses the same parameters. Tangent has no amplitude because it is unbounded; its basic period becomes $\frac{\pi }{\left|B\right|}$. These transformations describe tides, alternating signals and rotating mechanisms, but a periodic fit is meaningful only when the process itself repeats.