Trigonometry is an essential component of mathematics that focuses on the relationships between the angles and sides of a triangle. The three key trigonometric ratios – sine, cosine, and tangent, abbreviated as Sin Cos Tan, respectively, play a crucial role, especially in the context of a right-angled triangle. This article will dive deep into the Sin Cos Tan values, their formulas, relevant tables, the associated trigonometric ratios, and practical examples.
An Introduction to Sin Cos Tan
In the realm of trigonometry, the abbreviations Sin, Cos, and Tan represent the three fundamental trigonometric ratios: Sine, Cosine, and Tangent. These ratios are primarily applied to study the relationship between the angles and the sides of a right-angled triangle. The ancient mathematician Pythagoras established connections between the sides of a right-angled triangle, and Hipparcus laid the foundations for the relationship between the sides and angles of a triangle using trigonometry.
What are Sin Cos Tan Values?
Sin, Cos, and Tan values are the trigonometric ratios that facilitate the study of relationships between angles and sides of a right-angle triangle. These values are not entirely dependent on the lengths of the sides but also on the angle formed between the two sides. By knowing the Sin, Cos, and Tan values, we can determine the missing sides or angles of a right-angled triangle.
Components of Right Angle Triangle
In a right-angle triangle, we have three main components:
- Hypotenuse: The longest side of the triangle, opposite to the right angle.
- Opposite Side: The side of the triangle that is directly opposite to the angle under consideration. This is also known as the ‘perpendicular’ side.
- Adjacent Side: The side of the triangle that lies next to or ‘touches’ the angle. This is also referred to as the ‘base’ of the triangle.
Sin Cos Tan Formulas
Let’s define the Sin, Cos, and Tan formulas using two of the three sides of a right-angled triangle.
- Sine (Sin) θ = Opposite/Hypotenuse
- Cosine (Cos) θ = Adjacent/Hypotenuse
- Tangent (Tan) θ = Opposite/Adjacent
In addition to these basic trigonometric ratios, there are three additional ratios known as cosecant, secant, and cotangent, which are the reciprocals of Sin, Cos, and Tan respectively.
Derivative of Sin Cos Tan
The derivative of Sin, Cos, and Tan functions are crucial in calculus.
- The derivative of Sin θ is Cos θ.
- The derivative of Cos θ is -Sin θ.
- The derivative of Tan θ is Sec² θ or 1/Cos² θ.
How to Find Sin Cos Tan Values?
To find the Sin, Cos, and Tan values, you need to understand the right-angled triangle’s sides and angles. The Sin value is calculated by the ratio of the length of the side that is opposite to the angle over the length of the hypotenuse. The Cos value is calculated by the ratio of the length of the adjacent side over the length of the hypotenuse. Lastly, the Tan value is calculated by the ratio of the length of the side opposite to the angle over the length of the adjacent side.
Sin Values
For Sine Values,
- sin 0° = √(0/4) = 0
- sin 30° = √(1/4) = ½
- sin 45° = √(2/4) = 1/√2
- sin 60° = √3/4 = √3/2
- sin 90° = √(4/4) = 1
Cos Values
For Cos Values,
- cos 0° = √(4/4) = 1
- cos 30° = √(3/4) = √3/2
- cos 45° = √(2/4) = 1/√2
- cos 60° = √(1/4) = 1/2
- cos 90° = √(0/4) = 0
Tan Values
For Tan Values,
- tan 0° = 0/1 = 0
- tan 30° = (1/2) / (√3/2) = 1/√3
- tan 45° = (1/√2) / (1/√2) = 1
- tan 60° = [(√3/2)/(½)] = √3
- tan 90° = 1/0 = ∞
Trigonometric Ratios
Trigonometric ratios are mathematical tools that allow us to relate the angles of a triangle to the lengths of its sides. The primary trigonometric ratios are Sin, Cos, and Tan, with each of them providing a ratio of two sides of a right-angled triangle. We also have three additional trigonometric ratios – Cosecant (cosec), Secant (sec), and Cotangent (cot), which are the reciprocals of Sin, Cos, and Tan, respectively.
Sin Cos Tan on Unit Circle
The unit circle is a useful tool to determine the Sin, Cos, and Tan values for any given angle. A unit circle is a circle with a radius of 1, centered at the origin (0,0) in the Cartesian coordinate system. For a point on the unit circle with coordinates (x,y), the Sin, Cos, and Tan ratios can be given as:
- Sin θ = y/1
- Cos θ = x/1
- Tan θ = y/x
Where θ is the angle the line that connects the point and the origin forms with the positive x-axis.
Sin Cos Tan Chart
The Sin Cos Tan values for different angles can be summarized into a table, often referred to as the Sin Cos Tan chart. This chart is a ready reckoner to quickly look up the values of Sin Cos and Tan for standard angles – 0°, 30°, 45°, 60°, and 90°.
| Angles (in degrees) | Sin θ | Cos θ | Tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | ∞ |
Reciprocal Identities
The reciprocal identities are trigonometric identities that relate the reciprocal trigonometric functions – Cosecant, Secant, and Cotangent with the primary trigonometric functions – Sin, Cos, and Tan. They can be expressed as:
- Cosec θ = 1/Sin θ = Hypotenuse/Opposite
- Sec θ = 1/Cos θ = Hypotenuse/Adjacent
- Cot θ = 1/Tan θ = Adjacent/Opposite
Pythagorean Identities
The Pythagorean identities are fundamental trigonometric identities derived from the Pythagorean theorem. They connect the squares of the Sin and Cos functions of an angle and are given by the following equations:
- Sin² θ + Cos² θ = 1
- Sec² θ – Tan² θ = 1
- Cosec² θ – Cot² θ = 1
Negative Angle Identity
The negative angle identities express the Sin, Cos, and Tan of a negative angle in terms of the Sin, Cos, and Tan of the positive angle. They are:
- Sin (– θ) = – Sin θ
- Cos (– θ) = Cos θ
- Tan (– θ) = – Tan θ
Application of Sin Cos Tan Values
The concepts of Sin, Cos, and Tan find extensive applications in various fields. They are used to measure heights and distances, in navigation, in science and engineering, in the construction of parallels and perpendiculars, inclinations, and heights. Flight engineers use them to determine speed, distance, and direction changes due to wind loads in flight aviation. Even in the field of computer graphics and gaming, these concepts are used to create three-dimensional effects.
Trigonometric Functions
Trigonometric functions, also known as circular functions, are mathematical functions that relate an angle of a right-angled triangle with ratios of two of its sides. The six fundamental trigonometric functions are: Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent.
| Trigonometric Functions | Formula of Trigonometric Functions |
|---|---|
| Sin θ | Opposite/Hypotenuse |
| Cos θ | Adjacent/Hypotenuse |
| Tan θ | Opposite/Adjacent |
| Cosec θ | Hypotenuse/Opposite |
| Sec θ | Hypotenuse/Adjacent |
| Cot θ | Adjacent/Opposite |
Solved Examples on Sin Cos Tan
Here are some solved examples to understand the application of the Sin Cos Tan formulas and values.
Example 1: Find the value of Sin θ, Cos θ, and Tan θ for a right-angled triangle where the opposite side is 6 units and the adjacent side is 8 units, and the hypotenuse is 10 units.
Solution: Sin θ = Opposite/Hypotenuse = 6/10 = 0.6 Cos θ = Adjacent/Hypotenuse = 8/10 = 0.8 Tan θ = Opposite/Adjacent = 6/8 = 0.75
Example 2: Solve the expression tan 60(sec 60/cosec 60).
Solution: Sec 60/cosec 60 = Sin 60/Cos 60 tan 60(sec 60/cosec 60) = tan 60(Sin 60/Cos 60) = tan 60 × tan 60= √3² = 3
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