What Are Sin, Cos, and Tan?
In any right triangle, pick one of the two acute angles and call it $\theta$. The three sides have specific names relative to $\theta$:
Opposite — the side across from angle $\theta$.
Adjacent — the side next to angle $\theta$ that is not the hypotenuse.
Hypotenuse — the longest side, opposite the right angle.
The three primary trigonometric ratios are:
$$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}$$
These ratios are constant for a given angle. Two right triangles with the same angle $\theta$ have the same sin, cos, tan — regardless of size.
What Is SOHCAHTOA?
SOHCAHTOA is the standard mnemonic for the three trig ratio definitions:
SOH — Sine = Opposite / Hypotenuse
CAH — Cosine = Adjacent / Hypotenuse
TOA — Tangent = Opposite / Adjacent
Read aloud as "soh-kah-toh-uh," it's one of the most-taught mnemonics in middle and high school math worldwide.
A visual alternative: sine and cosine both have hypotenuse in the denominator — they relate the slanted side to the others. Tangent doesn't include the hypotenuse — it's the ratio of the two legs.
How Do You Use Sin Cos Tan?
Worked Example 1: Finding a Side
A right triangle has angle $\theta = 30°$ and hypotenuse 10. Find the opposite side.
$$\sin 30° = \frac{\text{opposite}}{10}$$
Since $\sin 30° = \tfrac{1}{2}$:
$$\frac{1}{2} = \frac{\text{opposite}}{10} \quad \Rightarrow \quad \text{opposite} = 5$$
Worked Example 2: Finding an Angle
A right triangle has opposite side 7 and adjacent side 24. Find the angle $\theta$.
$$\tan \theta = \frac{7}{24}, \quad \theta = \arctan!\left(\frac{7}{24}\right) \approx 16.26°$$
Worked Example 3: 5–12–13 Right Triangle
A right triangle has hypotenuse 13 and one leg 5. By Pythagoras the other leg is 12. Then $\sin\theta = 5/13 \approx 0.385$, so $\theta \approx 22.62°$. This is the famous 5–12–13 Pythagorean triple.
How Do Sin, Cos, Tan Relate to the Unit Circle?
The right-triangle definition only works for acute angles. To extend sin, cos, tan to any angle, we use the unit circle — a circle of radius 1 centred at the origin.
For an angle $\theta$ measured counterclockwise from the positive x-axis, the point on the unit circle is $(\cos\theta, \sin\theta)$. From this:
$\sin\theta = y$-coordinate of the point
$\cos\theta = x$-coordinate of the point
$\tan\theta = y/x$ — the slope of the line from the origin to the point
This extends sin and cos to all real numbers — including negative angles and angles greater than 360°. The unit-circle approach is what makes trigonometry work for periodic phenomena like waves and oscillations.
What Are the Standard Values of Sin Cos Tan?
Function | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
sin | 0 | $\tfrac{1}{2}$ | $\tfrac{1}{\sqrt{2}}$ | $\tfrac{\sqrt{3}}{2}$ | 1 |
cos | 1 | $\tfrac{\sqrt{3}}{2}$ | $\tfrac{1}{\sqrt{2}}$ | $\tfrac{1}{2}$ | 0 |
tan | 0 | $\tfrac{1}{\sqrt{3}}$ | 1 | $\sqrt{3}$ | Undefined |
Why Are Sin, Cos, Tan Important? (The Real-World GROUND)
"The heavens are like an open book, written in the language of mathematics." — adapted from Galileo, 1623.
Trigonometric ratios are 2,000 years old. Hipparchus of Nicaea (c. 190–c. 120 BCE) developed the first systematic trig table for astronomy. The Indian mathematician Aryabhata (476–550 CE) refined sine values to high accuracy and gave us the word sine itself — through a chain of translations from his Sanskrit jya via Arabic jiba and Latin sinus.
The reason these ratios appear in nearly every quantitative field:
Navigation. Latitude and longitude calculations use sin and cos directly. Marine navigation, aircraft navigation, and modern GPS all rely on trigonometric ratios.
Surveying. Land surveyors use trig to compute distances and elevations across terrain.
Engineering — structural analysis. Forces in trusses, stresses in beams, electrical phase angles in AC circuits — all expressed using sin and cos.
Physics — waves. Sound, light, water waves, and quantum mechanics all use sin/cos as the basic waveform.
Audio synthesis. Every digital audio sample is a sum of sines and cosines (Fourier series).
Computer graphics. 3D rotations multiply position vectors by matrices containing sin and cos.
Signal processing. Image compression (JPEG), data compression, and noise reduction all use the discrete cosine transform — built on cosine.
The modern symbolic notation — $\sin$, $\cos$, $\tan$ — was standardised by Leonhard Euler in his 1748 Introductio in Analysin Infinitorum.
A Worked Example
Find $\sin\theta$ if the adjacent side is 8 and the hypotenuse is 10.
The intuitive (wrong) approach. A student in a hurry plugs the adjacent into the sin formula directly:
$$\sin\theta \stackrel{?}{=} \frac{8}{10} = 0.8$$
Why it fails. Sin uses opposite over hypotenuse, not adjacent over hypotenuse. The formula being used is actually cosine.
The correct method.
Step 1: Find the opposite side using Pythagoras. $10^2 = 8^2 + \text{opp}^2$, so $\text{opp} = \sqrt{100 - 64} = 6$.
Step 2: Apply $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$:
$$\sin\theta = \frac{6}{10} = 0.6$$
Check. This is the 3-4-5 triangle scaled by 2 — sides 6, 8, 10. $\sin\theta = 0.6$, $\cos\theta = 0.8$, $\tan\theta = 0.75$. ✓
At Bhanzu, our trainers walk through this wrong-path-first sequence intentionally — confusing which side goes where is the most common archetype. Once the student feels the consequence of using adjacent in the sin formula, SOHCAHTOA stops being a mnemonic and becomes a structural rule.
What Are the Most Common Mistakes With Sin Cos Tan?
Mistake 1: Confusing opposite and adjacent
Where it slips in: Reading the wrong side as "opposite" or "adjacent" relative to angle $\theta$.
Don't do this: Using the side touching the angle as the "opposite" side.
The correct way: Opposite means the side directly across the triangle from $\theta$ — not touching $\theta$. Adjacent is the leg that touches $\theta$ (excluding the hypotenuse). Always identify $\theta$ first, then label sides.
Mistake 2: Using the wrong angle's perspective
Where it slips in: A right triangle has two acute angles. The "opposite" and "adjacent" sides switch depending on which acute angle you pick.
Don't do this: Computing sin of one angle but using the opposite/adjacent of the other acute angle.
The correct way: Pick one acute angle as $\theta$ and label sides relative to that angle. If you switch angles, the labels swap.
Mistake 3: Forgetting calculator mode (degrees vs radians)
Where it slips in: Computing $\sin(30)$ on a calculator set to radians, getting $-0.988$ instead of $0.5$.
Don't do this: Plugging $\sin(30)$ blindly without checking the mode.
The correct way: Always check whether your calculator is in DEG (degrees) or RAD (radians) mode. $\sin(30°) = 0.5$. $\sin(30\text{ rad}) \approx -0.988$. Many wrong physics-and-engineering answers come from this exact mode confusion.
The Mathematicians Who Shaped Sin Cos Tan
Hipparchus of Nicaea (c. 190–c. 120 BCE, Greece) — Built the first systematic trigonometric table around 130 BCE for astronomy. Often called the father of trigonometry.
Aryabhata (476–550 CE, India) — Refined sine values to high accuracy in his Aryabhatiya. The word sine comes from his Sanskrit jya through Arabic jiba and Latin sinus.
Leonhard Euler (1707–1783, Switzerland) — Standardised the modern notation $\sin$, $\cos$, $\tan$ in his 1748 Introductio in Analysin Infinitorum. Also unified trigonometry with complex numbers via Euler's identity.
A Practical Next Step
Try these three before moving on to inverse trig functions.
A right triangle has opposite = 3 and hypotenuse = 5. Find $\sin\theta$ and $\cos\theta$.
Find $\tan 60°$. Then find $\sin 60° / \cos 60°$ — do they match?
A right triangle has angle $\theta = 45°$ and hypotenuse = $10\sqrt{2}$. Find both legs.
Frequently Asked Questions
Q: What is SOHCAHTOA?
A memory mnemonic for the three trig ratios: SOH (sin = opposite/hypotenuse), CAH (cos = adjacent/hypotenuse), TOA (tan = opposite/adjacent).
Q: What is the difference between sin, cos, and tan?
Sin = opposite/hypotenuse. Cos = adjacent/hypotenuse. Tan = opposite/adjacent. All three are ratios of right-triangle sides. Sin and cos involve the hypotenuse; tan does not.
Q: What is sin 30°?
$\sin 30° = \tfrac{1}{2}$. From the 30-60-90 right triangle: the side opposite the 30° angle is half the hypotenuse.
Q: What is the value of tan 45°?
$\tan 45° = 1$. From the 45-45-90 isosceles right triangle: the opposite and adjacent legs are equal.
Q: How are sin, cos, tan related?
$\tan\theta = \frac{\sin\theta}{\cos\theta}$. Also: $\sin^2\theta + \cos^2\theta = 1$ (the Pythagorean identity). These two relationships connect all three ratios.
Q: When do we use sin, cos, or tan?
Use sin or cos when one side you know or want is the hypotenuse. Use tan when both known/wanted sides are legs (no hypotenuse involved). For finding an angle when you know two sides, use $\arcsin$, $\arccos$, or $\arctan$.
Q: How do you compute sin cos tan on a calculator?
Make sure the calculator is in the correct mode — DEG (degrees) for school problems, RAD (radians) for physics/calculus. Then type the function followed by the angle: $\sin(30)$ gives $0.5$ in degree mode.
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