Trigonometric Functions — Complete Guide to sin, cos, and tan Formulas

Trigonometric functions connect the size of an angle to the ratio of the sides of a triangle. Whenever a problem ties an angle to a height, distance, slope, or rotation, these three ratios are usually the entry point. The first three formulas students meet are the place to start.

The Formulas And What Each Symbol Means

For angle $\theta$ in a right triangle:

$\sin \theta = \frac{\text{맞은편}}{\text{빗변}}, \quad \cos \theta = \frac{\text{이웃변}}{\text{빗변}}, \quad \tan \theta = \frac{\text{맞은편}}{\text{이웃변}}$

Each ratio compares two sides named relative to angle $\theta$:

• Opposite (맞은편): the side directly across from angle $\theta$.
• Adjacent (이웃변): the side next to angle $\theta$ that is not the hypotenuse.
• Hypotenuse (빗변): the longest side, opposite the right angle.

The criteria for choosing a formula is simple:

• If you need the opposite and the hypotenuse $\rightarrow$ $\sin$
• If you need the adjacent and the hypotenuse $\rightarrow$ $\cos$
• If you need the opposite and the adjacent $\rightarrow$ $\tan$

And if $\cos \theta \ne 0$, you can use:

$\tan \theta = \frac{\sin \theta}{\cos \theta}$

If you can accurately grasp what these three ratios are comparing, you've already understood half of trigonometry.

Why These Ratios Stay Fixed

The core idea of trigonometry is that in right triangles with the same angle, the ratio of the sides remains constant. Even if the size of the triangle changes, as long as the angle is the same, the values of $\sin \theta$, $\cos \theta$, and $\tan \theta$ do not change.

Because of this, trigonometric functions are often used as a "way to find lengths using angles when it's difficult to measure the length directly." That is also why the side names are not fixed labels: even in the same triangle, if you change the reference angle, "opposite" and "adjacent" can switch.

Worked Example: Find sin, cos, and tan From Side Lengths

Consider a right triangle where the hypotenuse is $10$ and the adjacent side is $8$, and let angle $\theta$ be the angle attached to the adjacent side $8$.

First, find the length of the opposite side. By the Pythagorean theorem:

$\text{맞은편}^2 = 10^2 - 8^2 = 100 - 64 = 36$

So the opposite side is $6$. Now plug these into the ratios:

$\sin \theta = \frac{6}{10} = \frac{3}{5}$

$\cos \theta = \frac{8}{10} = \frac{4}{5}$

$\tan \theta = \frac{6}{8} = \frac{3}{4}$

The important part isn't the calculation itself, but the order: pick the reference angle, name the sides, and choose the required ratio. The formula follows naturally.

Now You Try

Change the numbers and run the same routine: in a right triangle, set the hypotenuse to $13$ and the adjacent side to $5$. Use the Pythagorean theorem to find the opposite side first, then read off $\sin \theta$, $\cos \theta$, and $\tan \theta$. If you can do this quickly, you've grasped the basic concepts; next, look at how $\sin$, $\cos$, and $\tan$ connect on the unit circle.

Calculation Traps To Watch

Naming sides before picking the reference angle

The opposite and adjacent sides change depending on the reference angle. If you don't check the angle first, you might use the correct formula but plug in the wrong sides.

Applying right triangle definitions to all triangles

The ratio definitions for $\sin$, $\cos$, and $\tan$ are specifically for right triangles. For triangles without a right angle, you usually need other tools like the Law of Sines or the Law of Cosines.

Forgetting to check the calculator's angle mode

If a problem gives angles in degrees, such as $30^\circ$ or $45^\circ$, your calculator must be in "degree" mode. If it's in "radian" mode, the values will be completely different.

Overlooking the conditions for tangent

Since $\tan \theta = \frac{\sin \theta}{\cos \theta}$, tangent is not defined where $\cos \theta = 0$. You must keep this condition in mind when transforming equations.

Moving Beyond Right Triangles: The Unit Circle

The right triangle definition is just the starting point. When an angle is greater than $90^\circ$ or is negative, we interpret trigonometric functions using the unit circle. If you consider a point corresponding to angle $\theta$ on a circle with radius $1$, the coordinates are:

$(\cos \theta, \sin \theta)$

So you can think of cosine as the x-coordinate and sine as the y-coordinate. Thanks to this expansion, trigonometry naturally leads into graphs, periodicity, and wave problems.

Where Is Trigonometry Used?

In school mathematics, it leads to right triangle problems, the unit circle, trigonometric graphs, and trigonometric equations. In physics, it's linked to waves and vibrations; in engineering, to slopes and rotations; and in coordinate analysis, to direction and distance problems. While it may seem like there are many formulas at first, it's actually just one sentence — "once an angle is fixed, the ratio of the sides is fixed" — expressed in several different ways.