Concept · Trigonometry
Sine, Cosine and Tangent: What They Are, Plainly
Sine, cosine and tangent are just three ratios of a triangle's sides. The unit circle extends them to every angle, and shows why the signs change.
Three ratios of one triangle
Take a right triangle and pick one of its acute angles. Relative to that angle, the three sides have names: the opposite side across from it, the adjacent side next to it, and the hypotenuse, the longest side facing the right angle. Sine, cosine and tangent are the three ratios you can make from them.
| Function | Ratio | Memory phrase |
|---|---|---|
| sine | opposite ÷ hypotenuse | SOH |
| cosine | adjacent ÷ hypotenuse | CAH |
| tangent | opposite ÷ adjacent | TOA |
That is where the phrase SOH CAH TOA comes from. The important fact is that each ratio depends only on the angle, not on the size of the triangle: a 30° angle always has a sine of 1/2, however large the triangle is drawn.
The unit circle: the same ideas for every angle
A triangle only holds angles below 90°. The unit circle removes that limit. Draw a circle of radius 1 centred at the origin, and turn an angle from the positive x-axis. The point where the angle meets the circle has coordinates (cos θ, sin θ).
So cosine is the x-coordinate and sine is the y-coordinate, for any angle at all. Tangent is then sin θ ÷ cos θ, the slope of the line from the origin to that point. Because the coordinates can be negative, so can sine and cosine, which is why the functions are not only positive.
- 150°an angle in the second quadrant, past 90°
- reference angle 30°the acute angle between the terminal side and the x-axis
- sin 150° = 1/2the same size as sin 30°; the y-coordinate is still above the axis, so positive
- cos 150° = −√3/2the same size as cos 30°; the x-coordinate is left of the axis, so negative
Signs by quadrant
Which functions are positive depends on the quadrant, because it decides the signs of x and y.
| Quadrant | Angles | Positive functions |
|---|---|---|
| I | 0° to 90° | sin, cos, tan |
| II | 90° to 180° | sin only |
| III | 180° to 270° | tan only |
| IV | 270° to 360° | cos only |
A reference angle gives the size of any value, and the quadrant gives its sign. That one idea is behind every trig table.
Cosecant, secant and cotangent
The other three functions are reciprocals: cosecant is 1 ÷ sine, secant is 1 ÷ cosine, and cotangent is 1 ÷ tangent, which is cosine ÷ sine. They matter because a reciprocal is undefined wherever the original function is zero.
When a value is undefined
Tangent is sine divided by cosine, and cosine is 0 at 90° and 270°. Division by zero has no answer, so tan 90° is undefined: not zero, and not a very large number. Graphically the tangent curve shoots upward without ever reaching a value there. The same holds for secant at those angles and for cosecant and cotangent at 0° and 180°.
A worked triangle
A right triangle with sides 3, 4 and 5 shows the ratios at work. The hypotenuse is 5. For the angle opposite the side of length 3, the opposite side is 3 and the adjacent side is 4.
| Ratio | Working | Value |
|---|---|---|
| sin | 3 ÷ 5 | 0.6 |
| cos | 4 ÷ 5 | 0.8 |
| tan | 3 ÷ 4 | 0.75 |
A quick check that these three belong together: 0.6² + 0.8² = 0.36 + 0.64 = 1, which is the Pythagorean theorem written for the unit circle. Doubling every side to 6, 8 and 10 changes none of the ratios, which is why the functions belong to the angle rather than to the triangle's size.
Degrees, radians, and the standard angles
Angles can be measured in degrees or in radians, and the same number means different angles in each. Exact values exist only for the standard angles, the multiples of 30° and 45° (π/6 and π/4 radians). Everything else is an irrational decimal, which a calculator rounds.
Questions about sine cosine and tangent
What are sine, cosine and tangent?
Three ratios of a right triangle's sides: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. On the unit circle, cosine and sine are the x- and y-coordinates.
What does SOH CAH TOA mean?
A memory aid: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Why is tan 90° undefined?
Because tan = sin ÷ cos and cos 90° = 0. Dividing by zero has no answer, so the value does not exist.
Why can sine and cosine be negative?
Because on the unit circle they are coordinates, and coordinates can be negative. The quadrant decides which of them is.