What is Sine, Cosine and Tangent?
Sine, cosine, and tangent (abbreviated as sin, cos, and tan) are three primary trigonometric functions, which relate an angle of a right-angled triangle to the ratios of two sides length. The reciprocals of sine, cosine, and tangent are the secant, the cosecant, and the cotangent respectively. Each of the six trigonometric functions has corresponding inverse functions (also known as inverse trigonometric functions). The trigonometric functions also known as the circular functions, angle functions, or goniometric functions are widely used in all fields of science that are related to Geometry such as navigation, celestial mechanics, solid mechanics, etc.
Read below to know what is a sine function, cosine function, and tangent function in detail.
Sine Cosine Tangent Definition
A right-angled triangle includes one angle of 90 degrees and two acute angles. Each acute angle of a right-angled triangle retains the property of the sine cosine tangent. The sine, cosine, and tangent of an acute angle of a right-angled triangle are defined as the ratio of two of three sides of the right-angled triangle.
As we know, sine, cosine, and tangent are based on the right-angled triangle, it would be beneficial to give names to each of the triangles to avoid confusion.
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“Hypotenuse side” is the longest side.
“Adjacent side” is the side next to angle θ.
“Opposite side” is the side opposite to angle θ.
Accordingly,
Sin θ = Opposite side/Hypotenuse
Cos θ = Adjacent/Hypotenuse
Tan θ = Opposite/Adjacent
What is the Sine Function?
In the right triangle, the sine function is defined as the ratio of the length of the opposite side to that of the hypotenuse side.
Sin θ = Opposite Side/ Hypotenuse Side.
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For example, the sine function of a triangle ABC with an angle θ is expressed as:
Sin θ = a/c
What is the Cosine Function?
In the right triangle, the cosine function is defined as the ratio of the length of the adjacent side to that of the hypotenuse side.
Cos θ = Adjacent Side/Hypotenuse Side
Example:
Considering the figure given above, the cosine function of a triangle ABC with an angle θ is expressed as:
Cos θ = b/c
What is the Tangent Function?
In the right triangle, the tangent function is defined as the ratio of the length of the opposite side to that of the adjacent side.
Tan θ = Opposite Side/Adjacent Side
Example:
Considering the figure given above, the cosine function of a triangle ABC with an angle θ is expressed as:
Tan θ = a/b
Sine Cosine Tangent Table
The values of trigonometric ratios like sine, cosine, and tangent for some standard angles such as 0°, 30°, 45°, 60°, and 90° can be easily determined with the help of the sine cosine tangent table given below. These values are very important to solve trigonometric problems. Hence, it is important to learn the values of trigonometric ratios of these standard angles.
The sine, cosine, and tangent table given below includes the values of standard angles like 0°, 30°, 45°, 60°, and 90°.
Sine, Cosine, and Tangent Table
Did You Know?
Sine and Cosine were introduced by Aryabhatta, whereas the tangent function was introduced by Muhammad Ibn Musa al- Khwarizmi ( 782 CE - 850 CE).
Sine Cosine and Tangent formulas can be easily learned using SOHCAHTOA. As sine is opposite side over hypotenuse side, cosine is adjacent side over hypotenuse side, and tangent is opposite side over the adjacent side.
Solved Examples:
1. Find Cos θ with respect to the following triangle.
Ans: To find Cos θ, we need both adjacent and hypotenuse side.
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The adjacent side in the above triangle is, BC = 8 Cm
But, the hypotenuse side i.e. AC is not given.
To find the hypotenuse side, we use the Pythagoras theorem
AC² = AB² + BC² = 6² + 8² = 100
Hypotenuse side, AC = √100 or 10 cm
Cos θ = Adjacent/Hypotenuse = 8/10
= 4/5
Therefore, Cos θ = 4/5
2. Find the value of Sin 45°, Cos 60°, and Tan 60°.
Solution: Using the trigonometric table above, we have:
Sin 45° = 1/√2
Cos 60° = 1/2
Tan 45°= 1
FAQs on Sine, Cosine and Tangent
1. What are sine, cosine, and tangent in the context of a right-angled triangle?
Sine, cosine, and tangent are the three primary trigonometric ratios that relate an angle in a right-angled triangle to the ratio of the lengths of its two sides. For a given angle θ:
- Sine (sin θ) is the ratio of the length of the opposite side to the length of the hypotenuse.
- Cosine (cos θ) is the ratio of the length of the adjacent side to the length of the hypotenuse.
- Tangent (tan θ) is the ratio of the length of the opposite side to the length of the adjacent side.
2. What is the easy way to remember the formulas for sine, cosine, and tangent?
A popular mnemonic to remember the trigonometric formulas is SOH CAH TOA. This stands for:
- SOH: Sine = Opposite / Hypotenuse
- CAH: Cosine = Adjacent / Hypotenuse
- TOA: Tangent = Opposite / Adjacent
By remembering this phrase, you can easily recall the correct ratio for each function.
3. How do you decide when to use sin, cos, or tan to solve a problem?
To decide which ratio to use, you should first identify the sides you know and the side you need to find, relative to the given angle (θ).
- Use sine (sin) when your problem involves the opposite side and the hypotenuse.
- Use cosine (cos) when your problem involves the adjacent side and the hypotenuse.
- Use tangent (tan) when your problem involves the opposite side and the adjacent side.
Choose the ratio that connects the known value with the unknown value you need to calculate.
4. What are the values of sin, cos, and tan for standard angles like 0°, 30°, 45°, 60°, and 90°?
The values of sine, cosine, and tangent for standard angles are fundamental in trigonometry. Key values as per the CBSE/NCERT curriculum for the 2025-26 session are:
- sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1
- cos 0° = 1, cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2, cos 90° = 0
- tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, tan 90° = Not Defined
5. What is the fundamental relationship between sine, cosine, and tangent?
The three ratios are fundamentally connected through two key identities. The most important relationship is that the tangent of an angle is the sine of the angle divided by its cosine: tan θ = sin θ / cos θ. Another critical relationship is the Pythagorean identity, which states that for any angle θ, sin²θ + cos²θ = 1. These identities are essential for simplifying expressions and solving trigonometric equations.
6. What are some real-world examples and applications of sine, cosine, and tangent?
Trigonometric functions are crucial in many fields beyond the classroom. Some important applications include:
- Architecture and Engineering: To calculate building heights, structural loads, and bridge spans.
- Navigation: Used in GPS, aviation, and marine navigation to pinpoint locations and plot courses.
- Physics: To describe sound and light waves, analyse oscillations, and understand projectile motion.
- Astronomy: To measure the distance to nearby stars and planets.
- Video Games and Computer Graphics: To create realistic 3D models and simulate object movement.
7. Why do the values of sine, cosine, and tangent depend only on the angle, not the size of the triangle?
This is because of the properties of similar triangles. All right-angled triangles that share a specific acute angle (e.g., 30°) are similar to each other. This means their corresponding sides are in proportion. While the absolute lengths of the sides will change with the triangle's size, the ratio of any two sides (like Opposite/Hypotenuse) will remain constant for that specific angle. Therefore, sin, cos, and tan are functions of the angle itself.
8. What are the reciprocal trigonometric ratios related to sin, cos, and tan?
For every primary trigonometric ratio, there is a corresponding reciprocal ratio. These are:
- Cosecant (csc θ) is the reciprocal of sine, so csc θ = 1 / sin θ = Hypotenuse / Opposite.
- Secant (sec θ) is the reciprocal of cosine, so sec θ = 1 / cos θ = Hypotenuse / Adjacent.
- Cotangent (cot θ) is the reciprocal of tangent, so cot θ = 1 / tan θ = Adjacent / Opposite.
