Trigonometry Table – Formula, Function, Identities, Graph & Examples

Ranjay Kumar

Trigonometry Table
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Trigonometry is a branch of mathematics that explores the relationships between the angles and sides of triangles. Central to this study is the trigonometry table, a valuable tool that lists the values of trigonometric functions such as sine, cosine, and tangent for various angles. These functions are fundamental in understanding the properties of right-angled triangles and the unit circle.

Formulas in trigonometry provide the mathematical framework to solve problems involving triangles and periodic phenomena. The primary trigonometric functions—sine, cosine, tangent, cotangent, secant, and cosecant—are defined through ratios of a triangle’s sides and extend to the unit circle for broader applications.

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the included variables. These identities, such as the Pythagorean identity and angle sum formulas, simplify complex expressions and solve equations more efficiently.

Graphs of trigonometric functions illustrate their periodic nature and key properties, such as amplitude, period, and phase shift. These visual representations are crucial in fields ranging from physics to engineering.

Examples and applications of trigonometry abound in real life, from calculating heights and distances to analyzing sound waves and electrical currents. Understanding the trigonometry table, along with its associated formulas, functions, identities, and graphs, is essential for mastering this versatile and practical area of mathematics.

A trigonometry table is an essential mathematical tool that provides the values of trigonometric functions for different angles. Trigonometry, a branch of mathematics, explores the relationships between the angles and sides of triangles. The primary trigonometric functions—sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot)—are crucial for solving problems related to angles and distances. These tables are particularly important for board exams and competitive exams.

Trigonometry Table

A trigonometry table provides the values of fundamental trigonometric functions such as sine (sin), cosine (cos), and tangent (tan) for commonly used angles, typically in degrees or radians. These tables are structured with angles like 0°, 30°, 45°, 60°, and 90°, allowing users to quickly reference the corresponding function values without needing to calculate them manually.

Angle (Degrees)30°45°60°90°
Angle (Radians)0π/6π/4π/3π/2
sin01/2√2/2√3/21
cos1√3/2√2/21/20
tan01/√31√3Undefined

The sine (sin) function represents the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle. The cosine (cos) function represents the ratio of the length of the adjacent side to the length of the hypotenuse. The tangent (tan) function represents the ratio of the length of the opposite side to the length of the adjacent side. These functions form a fundamental table in trigonometry, crucial for solving problems involving right triangles and applicable across physics, engineering, and mathematics.

Trigonometric Table

Trigonometric tables today may extend beyond standard functions to include hyperbolic trigonometric values, which are derived from hyperbolas rather than circles.

With the widespread availability of digital tools, trigonometric functions in tables can now be swiftly computed using calculators, computers, and various software applications. These resources ensure precise and reliable calculations for trigonometric functions across all angles, whether in degrees or radians.

Trigonometry Table Values

Degrees (°)Sine (sin)Cosine (cos)Tangent (tan)Cosecant (csc)Secant (sec)Cotangent (cot)
010undefined1undefined
30°1/2√3/2√3/32√3/32√3
45°√2/2√2/21√2√21
60°√3/21/2√32/√321/√3
90°10undefined1undefined0
120°√3/2-1/2-√3-2/√3-2-1/√3
135°√2/2-√2/2-1-√2-√2-1
150°1/2-√3/2-√3/3-2√3/3-2-√3
180°0-10undefined-1undefined
210°-1/2-√3/2√3/3-2√3/3-2√3
225°-√2/2-√2/21-√2-√21
240°-√3/2-1/2√3-2/√3-21/√3
270°-10undefined-1undefined0
300°-√3/21/2-√32/√32-1/√3
315°-√2/2√2/2-1√2√2-1
330°-1/2√3/2-√3/32√3/32√3
360°010undefined1undefined

Please note: In this table, “undefined” denotes that the trigonometry formula is not applicable for that specific angle. Additionally, the values displayed are rounded to several decimal places for simplicity.

Sin Cos Table
Angles (In Degrees) 30° 45° 60° 90° 180° 270° 360°
Angles (In Radians) π/6 π/4 π/3 π/2 π 3π/2
sin 0 1/2 1/√2 √3/2 1 0 -1 0
cos 1 √3/2 1/√2 1/2 0 -1 0 1

Trigonometry Table for Class 10

Trigonometry is introduced to students in class 10 across most Indian education boards, often posing a challenge to those entering their final year of high school. This branch of mathematics appears daunting at first due to its unfamiliarity. To aid students in grasping these concepts, a trigonometry table has been devised specifically for standard angles: 0°, 30°, 45°, 60°, and 90°. This table provides the values of Sin and Cos for these angles, crucial for understanding and mastering trigonometric principles. Memorizing this table not only facilitates success in school examinations but also establishes a solid groundwork for advanced studies. Below is the trigonometry table tailored for class 10:

Angle Sin Value Cos Value
0 1
30° 1/2 √3/2
45° √2/2 √2/2
60° √3/2 1/2
90° 1 0

In exam situations, students often face nervousness which can cause them to forget memorized information. To counter this, a helpful technique for learning trigonometry tables comes into play. This trick allows you to derive trigonometric ratios without needing to memorize them outright.

Instead of rote memorization, follow these steps to find the values for standard angles:

  • List the standard angles from 0 to 90 degrees.
  • Write whole numbers starting from 0 beneath these angles: 0 for 0 degrees, 1 for 30 degrees, and so forth.
  • Divide these numbers by 4, the largest value.
  • Take the square root of the divided results.
  • These results give you the values of sine (Sin) for the standard angles.
  • Write down these sine values in reverse order to obtain the cosine (Cos) values.

With sine and cosine values at your fingertips, you can easily determine the values of other trigonometric ratios. This method ensures you can calculate trigonometric values confidently, even if you forget the exact trigonometry table during your exam.

Trigonometric Functions Table

The sine (sin), cosine (cos), and tangent (tan) are essential trigonometric functions used in mathematics and science. A standard table for these functions provides their values at significant angles, typically in degrees and sometimes in radians, within a right-angled triangle. Below is a commonly referenced table showing sin, cos, and tan values at angles of 0°, 30°, 45°, 60°, and 90°:

Angle (°)SinCosTan
0010
301/2√3/21/√3
45√2/2√2/21
60√3/21/2√3
9010Undefined

These values originate from the characteristics of right-angled triangles and the unit circle. They form the basis of trigonometry and find extensive application in mathematics, engineering, and physics. It’s crucial to recognize that the tangent function becomes undefined at 90° due to division by zero (tan θ = sin θ / cos θ, and cos 90° = 0).

Trigonometric Ratio Table

Trigonometric tables encompass essential ratios: sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot). These tables serve as fundamental tools in solving trigonometric problems, providing values for standard angles. Notably, Fast Fourier Transform algorithms also rely on these tables. This article explores effective methods for constructing trigonometric tables. Keep this page bookmarked for ongoing updates.

Trigonometry Ratio Table

Trigonometry Ratio Table
Angles (In Degrees)30°45°60°90°180°270°360°
Angles (In Radians)π/6π/4π/3π/2π3π/2
sin01/21/√2√3/210-10
cos1√3/21/√21/20-101
tan01/√31√300
cot√311/√300
cosec2√22/√31-1
sec12/√3√22-11

Trigo Table

Understanding the trigonometric ratios is crucial for students as it simplifies learning the Trigonometry Table. These ratios depend on specific trigonometric values. To facilitate understanding, we provide both the trigonometric table and its values below. Students are encouraged to familiarize themselves with the ratios and their relationships before delving into the steps of memorizing the trigonometric table. Familiarity with these ratios beforehand will significantly ease the process of learning trigonometry for students.

Trigonometry Table 0-360 Value

  1. sin x = cos (90° – x)
  2. cot x = tan (90° – x)
  3. sec x = cosec (90° – x)
  4. cos x = sin (90° – x)
  5. tan x = cot (90° – x)
  6. cosec x = sec (90° – x)
  7. 1/sin x = cosec x
  8. 1/tan x = cot x
  9. 1/cos x = sec x

Trigonometry Table Tricks To Learn Trigonometric Table

Learning the trigonometric table can be quite manageable, especially with the help of mnemonic devices and visualization techniques. Here’s how you can approach it:

Mnemonic Devices:

Mnemonics such as “Some People Have Curly Brown Hair” are great for quickly recalling sine values like 0, 1/2, √2/2, √3/2, and 1 for angles 0°, 30°, 45°, 60°, and 90° respectively.

Visualize the Unit Circle:

Understanding the unit circle conceptually maps trigonometric values as coordinates on a circle. This visualization aids in remembering sine and cosine values for common angles.

Relate to Special Triangles:

Memorize the properties of 30°-60°-90° and 45°-45°-90° triangles. These triangles directly correlate with sine, cosine, and tangent values, simplifying memorization.

Regular Practice:

Consistent practice through exercises and quizzes reinforces memory retention. This practice is crucial for quick recall during exams or practical applications.

Choosing the best method depends on your learning style, but combining these approaches can significantly enhance your understanding and memorization of the trigonometric table.

How to Create a Trigonometry Table?

To create a trigonometric table, follow these steps:

Step 1: Construct the table with angles (0°, 30°, 45°, 60°, 90°) in the top row and trigonometric functions (sin, cos, tan, cot, sec, cosec) in the first column.

Step 2: Calculate the sine values for the angles:

For 0° to 90°:

0° ⟶ sin 0° = 0
30° ⟶ sin 30° = 1/2
45° ⟶ sin 45° = 1/√2
60° ⟶ sin 60° = √3/2
90° ⟶ sin 90° = 1
For angles beyond 90°:

Use the identities:

sin (180° − x) = sin x
sin (180° + x) = -sin x
sin (360° − x) = -sin x

Calculate:

sin 180° = sin 0° = 0
sin 270° = -sin 90° = -1
sin 360° = sin 0° = 0

Angles (in Degrees) 30° 45° 60° 90° 180° 270° 360°
sin 0 1/2 1/√2 √3/2 1 0 -1 0


Step 3: To find the value of cosine (cos x), use the identity sin (90° – x) = cos x. Apply this to calculate cos x for different angles:

Example:

For x = 0°:
cos 0° = sin (90° – 0°) = sin 90°

For x = 30°:
cos 30° = sin (90° – 30°) = sin 60°

For x = 45°:
cos 45° = sin (90° – 45°) = sin 45°

For x = 60°:
cos 60° = sin (90° – 60°) = sin 30°

For x = 90°:
cos 90° = sin (90° – 90°) = sin 0°

For x = 180°:
cos 180° = sin (90° – 180°) = -sin 90°

For x = 270°:
cos 270° = sin (90° – 270°) = -sin 180°

For x = 360°:
cos 360° = sin (90° – 360°) = -sin 270°

Angles (in Degrees) 30° 45° 60° 90° 180° 270° 360°
cos 1 √3/2 1/√2 1/2 0 -1 0 1

To find the tangent (tan) of an angle, use the formula:

tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x}

Calculate the tangent values for each angle provided using this formula.

Angles (Degrees) 30° 45° 60° 90° 180° 270° 360°
tan 0 13\frac{1}{\sqrt{3}} 1 3\sqrt{3} ∞\infty 0 ∞\infty 0

Step 5: Calculate the values of cotangent (cot) for all angles.

The cotangent (cot x) is calculated as cot⁡x=1tan⁡x\cot x = \frac{1}{\tan x}.

Calculate the values of cot by substituting each angle:

Angles (In Degrees) 30° 45° 60° 90° 180° 270° 360°
cot √3 1 1/√3 0 0

Step 6: Calculate the values of cosecant (cosec) for all angles.

The cosecant (cosec x) is calculated as csc⁡x=1sin⁡x\csc x = \frac{1}{\sin x}.

Calculate the values of cosec by substituting each angle:

Angles (In Degrees) 30° 45° 60° 90° 180° 270° 360°
cosec 2 √2 2/√3 1 -1

Step 7: Calculate the values of secant (sec) for all angles.

The secant (sec x) is calculated as sec⁡x=1cos⁡x\sec x = \frac{1}{\cos x}.

Calculate the values of sec by substituting each angle:

Angles (In Degrees) 30° 45° 60° 90° 180° 270° 360°
sec 1 2/√3 √2 2 -1 1

Trigonometry Table in π

Check the full table listed below for trigonometric values in terms of π (pi).

Angles (In π) π/6 π/4 π/3 π/2 π 3π/2
sin 0 1/2 1/√2 √3/2 1 0 -1 0
cos 1 √3/2 1/√2 1/2 0 -1 0 1
tan 0 1/√3 1 √3 0 0
cot √3 1 1/√3 0 0
cosec 2 √2 2/√3 1 -1
sec 1 2/√3 √2 2 -1 1

Trigonometric Table Examples

  1. Finding Sin 15°: Sin 15° can be written as sin(45° – 30°).

    sin⁡15°=sin⁡(45°−30°)=sin⁡45°cos⁡30°−cos⁡45°sin⁡30°\sin 15° = \sin(45° – 30°) = \sin 45° \cos 30° – \cos 45° \sin 30°

    Using the trigonometric table values:

    sin⁡15°=(12⋅32)−(12⋅12)\sin 15° = \left(\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}\right) – \left(\frac{1}{2} \cdot \frac{1}{\sqrt{2}}\right)

    Simplifying gives:

    sin⁡15°=3−122\sin 15° = \frac{\sqrt{3} – 1}{2\sqrt{2}}
  2. Proving Cos 2θ = cos²θ – sin²θ for θ = 30°:

    R.H.S.=cos⁡230°−sin⁡230°=(12)2−(32)2=34−14=12\text{R.H.S.} = \cos² 30° – \sin² 30° = \left(\frac{1}{2}\right)² – \left(\frac{\sqrt{3}}{2}\right)² = \frac{3}{4} – \frac{1}{4} = \frac{1}{2} L.H.S.=cos⁡2⋅30°=cos⁡60°=12\text{L.H.S.} = \cos 2 \cdot 30° = \cos 60° = \frac{1}{2}

    Hence, L.H.S. = R.H.S. (Proved)

  3. Finding the length of a ladder: Given an angle of 60° and the base of the ladder 12.4 m away from the wall:

    cos⁡60°=12.4h⇒12=12.4h⇒h=2⋅12.4=24.8 m\cos 60° = \frac{12.4}{h} \Rightarrow \frac{1}{2} = \frac{12.4}{h} \Rightarrow h = 2 \cdot 12.4 = 24.8 \text{ m}

    Therefore, the length of the ladder is 24.8 m.

Trigonometry Table Practice Questions

  1. Using the Trigonometry Table, find the value of [43tan⁡260°+3cos⁡230°−2sec⁡230°−34cot⁡260°]\left[\frac{4}{3} \tan² 60° + 3 \cos² 30° – 2 \sec² 30° – \frac{3}{4} \cot² 60°\right].

  2. Prove that sin⁡230°+cos⁡230°=1\sin² 30° + \cos² 30° = 1 using Trigonometry Table values.

  3. Using the Trigonometric Table, find the values of (a) tan⁡(π4)\tan \left(\frac{\pi}{4}\right) (b) sec⁡(π6)\sec \left(\frac{\pi}{6}\right) (c) sec⁡(π3)\sec \left(\frac{\pi}{3}\right).

  4. If the length of the shadow of a tree is 3\sqrt{3} times its real height, calculate the angle of elevation of the sun.

Frequently Asked Questions

What is a Trigonometry Table?

A Trigonometry Table lists values of trigonometric functions (sin, cos, tan, etc.) for standard angles like 0°, 30°, 45°, 60°, and 90°, often in degrees or radians.

What are the basic trigonometric functions?

The basic trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (cosec or csc), secant (sec), and cotangent (cot).

How do you use a Trigonometry Table to find values?

To find a trigonometric function value for an angle, locate the angle in degrees or radians on the table and read the corresponding value for the function.

What are some common trigonometric identities?

Common trigonometric identities include Pythagorean identities (sin²θ + cos²θ = 1), sum and difference identities (sin(A ± B) = sinA cosB ± cosA sinB), and double angle identities (sin(2θ) = 2sinθ cosθ).

How do you solve trigonometric equations using identities?

Trigonometric equations can be solved by manipulating them using trigonometric identities to simplify and isolate the variable.

What are the applications of trigonometric functions in real life?

Trigonometric functions are used in fields such as engineering (for designing structures), physics (in analyzing motion), astronomy (to determine positions of celestial bodies), and more.

How are trigonometric functions represented on a graph?

Sin, cos, tan, and other trigonometric functions are plotted on Cartesian coordinate systems where the x-axis represents angles and the y-axis represents function values.

What are the properties of trigonometric functions?

Trigonometric functions have periodicity (repeat values at regular intervals), symmetry (even or odd functions), and specific ranges of values depending on the function.

How do trigonometric functions relate to right-angled triangles?

In right-angled triangles, trigonometric functions relate ratios of sides (opposite, adjacent, and hypotenuse) to angles, helping calculate unknown sides or angles.

Can you give examples of using trigonometric functions?

Examples include finding heights using angles of elevation or depression, analyzing oscillatory motion using sine functions, and calculating forces in mechanical systems using cosine functions.

conclusion

Trigonometry Tables serve as indispensable tools in mathematics, offering a comprehensive reference for trigonometric functions across standard angles. They provide precise values for sine, cosine, tangent, and their reciprocal functions, facilitating calculations in fields ranging from engineering and physics to astronomy and beyond.

Understanding trigonometric formulas, such as Pythagorean identities and sum/difference identities, enhances problem-solving capabilities by simplifying equations and relationships involving angles. These identities, combined with the graphical representation of trigonometric functions on Cartesian planes, illustrate their periodic nature and essential properties, aiding in visualizing and analyzing various phenomena.

Real-world applications demonstrate the practical utility of trigonometry, from determining distances and angles in surveying and navigation to modeling oscillatory behavior in physics and engineering. By leveraging trigonometric identities and functions, one can derive solutions to complex problems involving angles, triangles, and periodic phenomena, thereby advancing scientific understanding and technological innovation.

In essence, Trigonometry Tables encapsulate the foundational principles, functions, identities, and graphical representations essential for mastering trigonometry. They provide a robust framework for solving diverse mathematical challenges and exploring the interconnectedness of angles, ratios, and real-world applications.

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