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

Trigonometric functions describe the relationship between angles and proportions. Most introductory problems start from three quantities: $\sin \theta$, $\cos \theta$, and $\tan \theta$. If you are looking for "trigonometric formulas for sin, cos, and tan," the right-triangle definitions plus one common identity are enough to solve many basic problems.

The Formulas And What Each Symbol Means

In a right triangle, the four most commonly used relationships are:

$\sin \theta = \frac{\text{对边}}{\text{斜边}}, \qquad \cos \theta = \frac{\text{邻边}}{\text{斜边}}, \qquad \tan \theta = \frac{\text{对边}}{\text{邻边}}$

plus one common relationship:

$\tan \theta = \frac{\sin \theta}{\cos \theta} \qquad (\cos \theta \ne 0)$

The "opposite side" (对边) and "adjacent side" (邻边) must always be determined relative to angle $\theta$. If the angle changes, the opposite and adjacent sides also switch — this is where beginners most often get confused. The "hypotenuse" (斜边) is the longest side, across from the right angle. Get comfortable with these four first before memorizing more complex variations; it's much more efficient.

Beyond the definitions, three groups of formulas are the most worth memorizing at the introductory stage:

Group 1: The Pythagorean Identity:

$\sin^2 \theta + \cos^2 \theta = 1$

Group 2: The relationship between tangent, sine, and cosine:

$\tan \theta = \frac{\sin \theta}{\cos \theta} \qquad (\cos \theta \ne 0)$

Group 3: Common values for special angles:

$\sin 30^\circ = \frac{1}{2}, \qquad \cos 30^\circ = \frac{\sqrt{3}}{2}, \qquad \tan 30^\circ = \frac{\sqrt{3}}{3}$

$\sin 45^\circ = \frac{\sqrt{2}}{2}, \qquad \cos 45^\circ = \frac{\sqrt{2}}{2}, \qquad \tan 45^\circ = 1$

$\sin 60^\circ = \frac{\sqrt{3}}{2}, \qquad \cos 60^\circ = \frac{1}{2}, \qquad \tan 60^\circ = \sqrt{3}$

Many basic problems eventually boil down to $30^\circ$, $45^\circ$, and $60^\circ$, so these values are worth memorizing directly.

Why These Ratios Hold

In a right triangle, once you fix an acute angle $\theta$, trigonometric functions describe the ratio between sides, not the length of a specific side itself. Because they are ratios, as long as the angle stays the same, the values of $\sin \theta$, $\cos \theta$, and $\tan \theta$ will not change, regardless of whether the triangle is scaled up or down.

If the problem discusses any arbitrary angle rather than just an acute angle, the more robust definition comes from the unit circle. The coordinates of the point on the unit circle corresponding to angle $\theta$ are $(\cos \theta, \sin \theta)$, which is why $\sin$ and $\cos$ aren't limited to right-triangle problems.

Worked Example: Find sin, cos, and tan Given Hypotenuse And Opposite Side

Suppose a right triangle has an acute angle $\theta$, a hypotenuse of length $10$, and an opposite side of length $6$. Find $\sin \theta$, $\cos \theta$, and $\tan \theta$.

First, find the most direct one:

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

To calculate $\cos \theta$ and $\tan \theta$, we still need the adjacent side. Since this is a right triangle, use the Pythagorean theorem first:

$\text{邻边} = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = 8$

Then plug it back into the definitions:

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

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

The key isn't the arithmetic, but the sequence: first fix the angle, then distinguish between the opposite and adjacent sides, and finally substitute according to the definitions. As long as this step is correct, trig problems usually won't get messy.

Now You Try

Run the same routine with new numbers: in a right triangle, take an acute angle $\theta$, set the hypotenuse as $13$ and the opposite side as $5$, then find $\sin \theta$, $\cos \theta$, and $\tan \theta$. Use the Pythagorean theorem for the adjacent side first. If you want to keep practicing, try a version where you are "given one angle and one side, and need to find the side length in reverse." Once you can link the definitions, the Pythagorean theorem, and result-checking together, you've truly mastered the basics.

Calculation Traps To Watch

Applying right-triangle definitions to all angles

The definition $\sin \theta = \frac{\text{对边}}{\text{斜边}}$ is suitable for acute angles in right triangles. For obtuse angles, negative angles, or angles greater than $360^\circ$, switch to the unit circle interpretation.

Forgetting the conditions for $\tan \theta$

Because

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

when $\cos \theta = 0$, $\tan \theta$ is undefined. This condition cannot be ignored, otherwise you might mistake an undefined value for a normal result.

Swapping the opposite and adjacent sides

The opposite and adjacent sides must be judged relative to the specified angle $\theta$; there is no single side on a diagram that is always called the "opposite side."

Memorizing values without checking for reasonableness

Within the range of acute angles, $\sin \theta$ and $\cos \theta$ are always positive and will not exceed $1$. If you calculate $\sin \theta = 1.4$, there is definitely an error in a previous step.

Where Are Trigonometric Functions Applied?

Trig functions most commonly appear in problems involving the sides and angles of right triangles, unit circles, wave and periodic models, coordinate decomposition, and later calculus. Whenever a problem involves angles, rotation, height, slope, or periodic changes, they will likely appear. If the core is "given one angle and one side, find another side," first think of $\sin$, $\cos$, and $\tan$; if the problem involves general angles, graphs, or identity transformations, switch to the unit circle and identities.