How to Remember and Use Sin, Cos, Tan Values in Trigonometry
The concept of sin cos tan values plays a key role in mathematics and is widely applicable in both real-life situations and competitive exam questions. Whether you're preparing for school exams, JEE, or just want to solve trigonometry problems faster, knowing these values and how to remember them is essential.
What Are Sin Cos Tan Values?
Sin cos tan values are the standard results of the basic trigonometric ratios — sine (sin), cosine (cos), and tangent (tan) — at commonly used angles such as 0°, 30°, 45°, 60°, and 90°. These values connect angles with sides of right-angled triangles and are used to solve height, distance, and other practical problems in Maths, Physics, and Engineering.
Key Formula for Sin Cos Tan
Here are the standard formulas that define these ratios in any right-angled triangle with an angle \( θ \):
- sin θ = Opposite Side / Hypotenuse
- cos θ = Adjacent Side / Hypotenuse
- tan θ = Opposite Side / Adjacent Side
Sin Cos Tan Values Table (0°, 30°, 45°, 60°, 90°)
Learn and use these sin, cos, tan values for quick reference during exams and problem solving. This printable chart is a must-memorize for all students.
| Angle (θ) | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | ∞ |
Sin Cos Tan Values in Radians
Many exams and higher studies use radians instead of degrees. Here are the same values in radians:
| Angle (θ) | 0 | π/6 | π/4 | π/3 | π/2 |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | ∞ |
Memory Trick: Sin Cos Tan Hand Rule
A simple way to remember the sin cos tan table is to use your left hand. For angles (0°, 30°, 45°, 60°, 90°), fold the finger that matches the angle (starting from the thumb as 0°). The number of fingers on the left = sin value; fingers on the right = cos value. Divide by 2 and take the square root as needed. The SOHCAHTOA rule is also useful: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
When to Use Sin, Cos, or Tan?
Use sin when you know or need the opposite and hypotenuse, cos for adjacent and hypotenuse, and tan for opposite and adjacent sides. These ratios help in solving triangles, heights and distances, and even engineering design problems. Students preparing for competitive exams like JEE and NEET can save a lot of time by instantly recalling these values.
Solved Example with Sin Cos Tan Values
Let's look at a typical exam question using these values:
Example: In triangle ABC, right-angled at B, with AB = 24 cm and BC = 7 cm, find sin A and cos A.
1. Use Pythagoras: AC² = AB² + BC² = 24² + 7² = 576 + 49 = 625
2. So, AC = 25
3. sin A = Opposite/Hypotenuse = BC/AC = 7/25
4. cos A = Adjacent/Hypotenuse = AB/AC = 24/25
Practice Questions: Check Your Sin Cos Tan Skills
- What is sin 60°?
- Find tan 45° using the value table.
- If cos A = 1/2, what is the angle A?
- Use the hand trick to find sin 30°.
- Calculate cos (90° – θ) if sin θ = 1/2.
Frequent Errors and Misunderstandings
- Swapping sine and cosine values for the same angle.
- Forgetting to rationalize denominators for tan/cot values.
- Applying the wrong ratio (using sin when adjacent side is needed).
Relation to Other Trigonometry Topics
Knowing sin cos tan values makes it easier to tackle trigonometric ratios, trignometric values table, and apply concepts in real-world applications. Once memorized, these values help solve tougher problems like identities, equations, and inverse trig functions.
We explored sin cos tan values — including key formulas, value tables, memory hacks, and solved examples. With continuous practice and smart tricks, you'll become fast and accurate in using these ratios in any trigonometry problem. For more guidance and expert tips, visit Vedantu’s Trigonometric Functions or attend a live session. Keep practicing!
Also Read: Trigonometric Ratios | SOHCAHTOA Explained | Trigonometric Functions
FAQs on Sin, Cos, Tan Values Chart and Formula Explained
1. What are the basic trigonometric ratios: sine (sin), cosine (cos), and tangent (tan)?
The basic trigonometric ratios define the relationship between the angles and sides of a right-angled triangle. For a given angle θ:
- Sine (sin θ) is the ratio of the length of the side opposite the angle to the length of the hypotenuse (Opposite/Hypotenuse).
- Cosine (cos θ) is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse (Adjacent/Hypotenuse).
- Tangent (tan θ) is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle (Opposite/Adjacent). It is also equal to sin θ / cos θ.
2. What are the standard values for sin, cos, and tan at 0°, 30°, 45°, 60°, and 90°?
The values of sin, cos, and tan for standard angles are fundamental in trigonometry. They are:
- sin 0° = 0, cos 0° = 1, tan 0° = 0
- sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3
- sin 45° = 1/√2, cos 45° = 1/√2, tan 45° = 1
- sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3
- sin 90° = 1, cos 90° = 0, tan 90° = Not Defined
3. Why is a right-angled triangle used as the basis for defining sin, cos, and tan?
A right-angled triangle is used because its properties provide a consistent and simple framework for relating angles to side ratios. The presence of a fixed 90° angle means that the other two angles must be acute and add up to 90°. This fixed structure ensures that for any given acute angle, the ratio of the side lengths remains constant, regardless of the size of the triangle. This consistency allows sin, cos, and tan to be defined as true functions of the angle itself.
4. Is there an easy trick to remember the definitions of sin, cos, and tan?
Yes, a popular mnemonic to remember the trigonometric ratios is SOH-CAH-TOA. It stands for:
- SOH: Sine is Opposite over Hypotenuse.
- CAH: Cosine is Adjacent over Hypotenuse.
- TOA: Tangent is Opposite over Adjacent.
By remembering this phrase, you can quickly recall the formula for each of the three basic ratios.
5. What are the reciprocal trigonometric ratios and how do they relate to sin, cos, and tan?
The three reciprocal trigonometric ratios are cosecant (csc), secant (sec), and cotangent (cot). They are defined as the reciprocals of sin, cos, and tan, respectively:
- Cosecant (csc θ) = 1 / sin θ = Hypotenuse / Opposite
- Secant (sec θ) = 1 / cos θ = Hypotenuse / Adjacent
- Cotangent (cot θ) = 1 / tan θ = Adjacent / Opposite
These ratios are important for solving a wider range of trigonometric problems and simplifying expressions.
6. How are the values of sine and cosine fundamentally related through complementary angles?
Sine and cosine are co-functions, meaning the sine of an angle is equal to the cosine of its complement, and vice-versa. In a right-angled triangle, if the two acute angles are A and B, then A + B = 90°. The side opposite angle A is the side adjacent to angle B. This leads to the relationship: sin(A) = cos(90° - A) and cos(A) = sin(90° - A). For example, this is why sin(30°) = 1/2 is the same value as cos(60°).
7. What are some real-life examples of how sin, cos, and tan values are applied?
Trigonometric values are crucial in many fields. For example:
- In architecture and engineering, they are used to calculate building heights, roof slopes, and structural loads.
- In astronomy, they help determine the distance to nearby stars and the positions of celestial bodies.
- In navigation (for ships and aircraft), they are used to calculate headings and pinpoint locations using GPS.
- In video game development and computer graphics, they are used to model light, perspective, and the rotation of objects in a 3D space.
8. How are sin, cos, and tan values determined for angles greater than 90 degrees?
For angles beyond 90°, trigonometric values are determined using the unit circle, which is a circle with a radius of 1 centred at the origin of a Cartesian plane. The plane is divided into four quadrants. The sign (+ or -) of sin, cos, and tan depends on which quadrant the angle falls into:
- Quadrant I (0°-90°): All are positive.
- Quadrant II (90°-180°): Only sin is positive.
- Quadrant III (180°-270°): Only tan is positive.
- Quadrant IV (270°-360°): Only cos is positive.
The actual value is found by relating the angle back to a corresponding acute angle in Quadrant I, known as the reference angle.
9. What is the importance of the Pythagorean identity sin²θ + cos²θ = 1?
The identity sin²θ + cos²θ = 1 is one of the most important formulas in trigonometry. Its importance lies in its ability to connect sine and cosine directly. Derived from the Pythagorean theorem in a unit circle, it allows you to find the value of sin θ if you know cos θ, or vice-versa, without needing to know the angle itself. This identity is fundamental for proving other trigonometric identities and simplifying complex expressions in calculus, physics, and engineering.
