Trigonometry ā€“ Definition, Formulas, Ratios, Identities , PDF

Trigonometry

What is Trigonometry?

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. It primarily deals with the sine, cosine, and tangent functions, which are ratios of sides of right triangles. These functions have profound applications in both theoretical and practical aspects, including physics, engineering, surveying, and computer graphics.

Trigonometric Ratios (Sin, Cos, Tan)

Trigonometric ratios are fundamental to the study of trigonometry and are used to relate the angles and sides of a right triangle. The three primary trigonometric ratios are sine, cosine, and tangent, commonly abbreviated as sin, cos, and tan, respectively. Hereā€™s a breakdown of each:

Sine (sin)

The sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse. It is defined as: sinā”(Īø)=opposite/hypotenuse

Cosine (cos)

The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. It is defined as: cosā”(Īø)=adjacent/hypotenuse

Tangent (tan)

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. It is defined as: tanā”(Īø)=opposite/adjacent

Six Trigonometric Functions

1. Sine (sin)

  • Definition: The ratio of the length of the opposite side to the hypotenuse in a right triangle.
  • Formula: sinā”(Īø)=opposite/hypotenuseā€‹

2. Cosine (cos)

  • Definition: The ratio of the length of the adjacent side to the hypotenuse.
  • Formula: cosā”(Īø)=adjacent/hypotenuseā€‹

3. Tangent (tan)

  • Definition: The ratio of the length of the opposite side to the adjacent side.
  • Formula: tanā”(Īø)=opposite/adjacentā€‹

4. Cotangent (cot)

  • Definition: The reciprocal of the tangent or the ratio of the length of the adjacent side to the opposite side.
  • Formula: cotā”(Īø)=1/tanā”(Īø)=adjacent/oppositeā€‹

5. Secant (sec)

  • Definition: The reciprocal of the cosine or the ratio of the hypotenuse to the adjacent side.
  • Formula: secā”(Īø)=1/cosā”(Īø)=hypotenuse/adjacentā€‹

6. Cosecant (csc)

  • Definition: The reciprocal of the sine or the ratio of the hypotenuse to the opposite side.
  • Formula: cscā”(Īø)=1/sinā”(Īø)=hypotenuse/opposite

Even and Odd Trigonometric Functions

Odd Trigonometric Functions: A trigonometric function is classified as odd if it satisfies the condition f(āˆ’x)=āˆ’f(x) This implies that the function is symmetric with respect to the origin, meaning that reflecting the function across both axes results in the same graph.

Even Trigonometric Functions: A trigonometric function is classified as even if it satisfies the condition f(āˆ’x)=f(x). This indicates that the function is symmetric about the y-axis, meaning that reflecting the function across the y-axis results in the same graph.

FunctionFormulaDescription
Sine (sin)(sin(-x) = -sin(x))Sine is an odd function.
Cosine (cos)(cos(-x) = cos(x))Cosine is an even function.
Tangent (tan)(tan(-x) = -tan(x))Tangent is an odd function.
Cosecant (csc)(csc(-x) = -csc(x))Cosecant is an odd function.
Secant (sec)(sec(-x) = sec(x))Secant is an even function.
Cotangent (cot)(cot(-x) = -cot(x))Cotangent is an odd function.

Trigonometry Angles

1.Angle Measurement

  • Degrees: One of the most common units for measuring angles. A full circle is divided into 360 degrees.
  • Radians: Another vital unit for measuring angles, particularly in higher mathematics and applications involving calculus. A full circle is 2Ļ€ radians. Radians are particularly useful because they simplify the integration and differentiation of trigonometric functions.

2. Standard Angles

  • Acute Angles: Angles less than 90 degrees (or Ļ€/2 radians).
  • Right Angles: Exactly 90 degrees (or Ļ€/2 radians).
  • Obtuse Angles: Greater than 90 degrees but less than 180 degrees (or between Ļ€/2 and Ļ€ radians).
  • Straight Angles: Exactly 180 degrees (or Ļ€ radians).
  • Reflex Angles: More than 180 degrees but less than 360 degrees (or between Ļ€ and 2Ļ€ radians).

3. Special Angles in Trigonometry

  • 30Ā° (or Ļ€/6), 45Ā° (or Ļ€/4), and 60Ā° (or Ļ€/3) Angles: These angles are commonly used in trigonometry because they correspond to specific, easy-to-remember values of sine, cosine, and tangent.
  • 0Ā° and 90Ā° (or 0 and Ļ€/2 radians): These are critical for defining the initial and maximal values of the trigonometric functions.

4. Negative Angles and Angle Cycles

  • Negative Angles: Rotating in the clockwise direction, contrary to the positive counter-clockwise standard.
  • Angle Cycles (Periodicity): Trigonometric functions are periodic, meaning they repeat their values in regular intervals. For instance, sine and cosine have a period of 2Ļ€ radians, which means they repeat every 2Ļ€ radians.

5. Angle Sum and Difference Identities

  • These identities allow for the calculation of the sine, cosine, and tangent of the sum or difference of two angles, facilitating the simplification of complex trigonometric expressions and proving useful in various applications such as signal processing.

6. Quadrants

  • The coordinate plane is divided into four quadrants by the x-axis and y-axis, which affect the sign (+/-) of the trigonometric functions depending on the quadrant an angle terminates in.

Trigonometry values Table

List of Trigonometry Formulas

Download All Trigonometry Formulas

Reciprocal Identities

Cosecant, secant, and cotangent are the reciprocals of the basic trigonometric ratios: sine, cosine, and tangent, respectively. These reciprocal identities are derived from the properties of a right-angled triangle and play a crucial role in trigonometry. They are often used to simplify and solve trigonometric problems. The formulas for these reciprocal trigonometric identities, which are essential for various calculations and transformations in trigonometry, are:

  • cosec Īø = 1/sin Īø; sin Īø = 1/cosec Īø
  • sec Īø = 1/cos Īø; cos Īø = 1/sec Īø
  • cot Īø = 1/tan Īø; tan Īø = 1/cot Īø

Pythagorean Identities

The Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse (ccc) is equal to the sum of the squares of the other two sides (aaa and bbb). Mathematically, this is expressed as: cĀ² = aĀ² + bĀ² ,Using this theorem, we can derive Pythagorean identities in trigonometry, which allow us to convert one trigonometric ratio into another. These identities are fundamental in simplifying and solving trigonometric equations.

  • sinĀ²Īø + cosĀ²Īø = 1
  • secĀ²Īø ā€“ tanĀ²Īø = 1
  • cscĀ²Īø ā€“ cotĀ²Īø = 1

Trigonometric Ratio Table

Periodicity Identities (in Radians):

These formulas are used to shift the angles by Ļ€/2, Ļ€, 2Ļ€, etc. They are also called co-function identities.

  • sin (Ļ€/2 ā€“ A) = cos A & cos (Ļ€/2 ā€“ A) = sin A
  • sin (Ļ€/2 + A) = cos A & cos (Ļ€/2 + A) = ā€“ sin A
  • sin (3Ļ€/2 ā€“ A)  = ā€“ cos A & cos (3Ļ€/2 ā€“ A)  = ā€“ sin A
  • sin (3Ļ€/2 + A) = ā€“ cos A & cos (3Ļ€/2 + A) = sin A
  • sin (Ļ€ ā€“ A) = sin A &  cos (Ļ€ ā€“ A) = ā€“ cos A
  • sin (Ļ€ + A) = ā€“ sin A & cos (Ļ€ + A) = ā€“ cos A
  • sin (2Ļ€ ā€“ A) = ā€“ sin A & cos (2Ļ€ ā€“ A) = cos A
  • sin (2Ļ€ + A) = sin A & cos (2Ļ€ + A) = cos A

Cofunction Identities (in Degrees):

  • sin(90Ā°āˆ’x) = cos x
  • cos(90Ā°āˆ’x) = sin x
  • tan(90Ā°āˆ’x) = cot x
  • cot(90Ā°āˆ’x) = tan x
  • sec(90Ā°āˆ’x) = cosec x
  • cosec(90Ā°āˆ’x) = sec x

Sum & Difference Identities

  • sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
  • cos(x+y) = cos(x)cos(y)ā€“sin(x)sin(y)
  • š‘”š‘Žš‘›(š‘„+š‘¦)= (š‘”š‘Žš‘› š‘„+š‘”š‘Žš‘›š‘¦) / (1āˆ’š‘”š‘Žš‘› š‘„.š‘”š‘Žš‘› š‘¦)
  • sin(xā€“y) = sin(x)cos(y)ā€“cos(x)sin(y)
  • cos(xā€“y) = cos(x)cos(y) + sin(x)sin(y)
  • tan(x-y) = (tan x ā€“ tan y) (1+tan x.tan y)

Triple Angle Identities

  • Sin 3x = 3sin x ā€“ 4sinĀ³x
  • Cos 3x = 4cosĀ³x-3cos x
  • Tan 3x = (3tanx ā€“ tanĀ³x)/(1-3tanĀ²x)

Inverse Trigonometry Formulas

  • sinā»Ā¹ (ā€“x) = ā€“ sinā»Ā¹ x
  • cosā»Ā¹ (ā€“x) = Ļ€ ā€“ cosā»Ā¹ x
  • tanā»Ā¹ (ā€“x) = ā€“ tanā»Ā¹ x
  • cosecā»Ā¹ (ā€“x) = ā€“ cosecā»Ā¹ x
  • secā»Ā¹ (ā€“x) = Ļ€ ā€“ secā»Ā¹ x
  • cotā»Ā¹ (ā€“x) = Ļ€ ā€“ cotā»Ā¹ x

Trigonometric Sign Functions

  • sin (-Īø) = āˆ’ sin Īø
  • cos (āˆ’Īø) = cos Īø
  • tan (āˆ’Īø) = āˆ’ tan Īø
  • cosec (āˆ’Īø) = āˆ’ cosec Īø
  • sec (āˆ’Īø) = sec Īø
  • cot (āˆ’Īø) = āˆ’ cot Īø

Trigonometry Formulas For Class 10

Trigonometric Sign Functions

  • sin (-Īø) = āˆ’ sin Īø
  • cos (āˆ’Īø) = cos Īø
  • tan (āˆ’Īø) = āˆ’ tan Īø
  • cosec (āˆ’Īø) = āˆ’ cosec Īø
  • sec (āˆ’Īø) = sec Īø
  • cot (āˆ’Īø) = āˆ’ cot Īø

Trigonometric Identities

  1. sinĀ²A + cotĀ²A = 1
  2. tanĀ²A + 1 = secĀ²A
  3. cotĀ²A + 1 = cosecĀ²A

Periodic Identities

  • sin(2nĻ€ + Īø ) = sin Īø
  • cos(2nĻ€ + Īø ) = cos Īø
  • tan(2nĻ€ + Īø ) = tan Īø
  • cot(2nĻ€ + Īø ) = cot Īø
  • sec(2nĻ€ + Īø ) = sec Īø
  • cosec(2nĻ€ + Īø ) = cosec Īø

Complementary Ratios

Quadrant I

  • sin(Ļ€/2 āˆ’ Īø) = cos Īø
  • cos(Ļ€/2 āˆ’ Īø) = sin Īø
  • tan(Ļ€/2 āˆ’ Īø) = cot Īø
  • cot(Ļ€/2 āˆ’ Īø) = tan Īø
  • sec(Ļ€/2 āˆ’ Īø) = cosec Īø
  • cosec(Ļ€/2 āˆ’ Īø) = sec Īø

Quadrant II

  • sin(Ļ€ āˆ’ Īø) = sin Īø
  • cos(Ļ€ āˆ’ Īø) = -cos Īø
  • tan(Ļ€ āˆ’ Īø) = -tan Īø
  • cot(Ļ€ āˆ’ Īø) = ā€“ cot Īø
  • sec(Ļ€ āˆ’ Īø) = -sec Īø
  • cosec(Ļ€ āˆ’ Īø) = cosec Īø

Quadrant III

  • sin(Ļ€ + Īø) = ā€“ sin Īø
  • cos(Ļ€ + Īø) = ā€“ cos Īø
  • tan(Ļ€ + Īø) = tan Īø
  • cot(Ļ€ + Īø) = cot Īø
  • sec(Ļ€ + Īø) = -sec Īø
  • cosec(Ļ€ + Īø) = -cosec Īø

Quadrant IV

  • sin(2Ļ€ āˆ’ Īø) = ā€“ sin Īø
  • cos(2Ļ€ āˆ’ Īø) = cos Īø
  • tan(2Ļ€ āˆ’ Īø) = ā€“ tan Īø
  • cot(2Ļ€ āˆ’ Īø) = ā€“ cot Īø
  • sec(2Ļ€ āˆ’ Īø) = sec Īø
  • cosec(2Ļ€ āˆ’ Īø) = -cosec Īø

Sum and Difference of Two Angles

  • sin (A + B) = sin A cos B + cos A sin B
  • sin (A āˆ’ B) = sin A cos B ā€“ cos A sin B
  • cos (A + B) = cos A cos B ā€“ sin A sin B
  • cos (A ā€“ B) = cos A cos B + sin A sin B
  • tan(A + B) = [(tan A + tan B)/(1 ā€“ tan A tan B)]
  • tan(A ā€“ B) = [(tan A ā€“ tan B)/(1 + tan A tan B)]

Double Angle Formulas

  • sin 2A = 2 sin A cos A = [2 tan A /(1 + tanĀ²A)]
  • cos 2A = cosĀ²A ā€“ sinĀ²A = 1 ā€“ 2 sinĀ²A = 2 cosĀ²A ā€“ 1 = [(1 ā€“ tanĀ²A)/(1 + tanĀ²A)]
  • tan 2A = (2 tan A)/(1 ā€“ tan2A)

Triple Angle Formulas

  • sin 3A = 3 sinA ā€“ 4 sinĀ³A
  • cos 3A = 4 cosĀ³A ā€“ 3 cos A
  • tan 3A = [3 tan A ā€“ tanĀ³A]/[1 āˆ’ 3 tanĀ²A]

FAQs

What Grade Level is Trigonometry?

Trigonometry is typically taught in high school, usually in the 10th or 11th grade, depending on the school curriculum and the studentā€™s track in mathematics.

What is the Basic Knowledge of Trigonometry?

Basic knowledge of trigonometry includes understanding angles, trigonometric ratios like sine, cosine, and tangent, and how to apply these to calculate distances and angles in triangles.

How to Do Trigonometry Easily?

To do trigonometry easily, familiarize yourself with the unit circle, common angle values, and trigonometric identities. Practice solving problems step-by-step and use visual aids to help understand concepts better.

Is Trigonometry a Hard Class?

Trigonometry can be challenging for some students due to its abstract concepts and the need to understand both geometric and algebraic representations of problems.

What is Trigonometry vs. Geometry?

Trigonometry focuses on the relationships within triangles, particularly right triangles, using angles and ratio functions. Geometry is broader, dealing with properties and relations of points, lines, surfaces, and solids.

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Practice Test

What is the value of sin 30 degrees?

0.5

0.6

0.7

1

of 10

What is the value of cos 90 degrees?

0

0.5

1

-1

of 10

What is the tangent of 45 degrees?

0

0.5

1

2

of 10

Which of the following is the Pythagorean identity?

sinĀ²Īø + cosĀ²Īø = 1

sinĀ²Īø + tanĀ²Īø = 1

cosĀ²Īø - sinĀ²Īø = 1

 tanĀ²Īø + 1 = sinĀ²Īø

of 10

What is the value of tan 30 degrees?

āˆš3

1

āˆš3/3

2

of 10

Which function represents the ratio of the length of the opposite side to the length of the adjacent side in a right triangle?

Cosine

Sine

Tangent

Secant

of 10

What is the value of sin(30Ā°)?

0

0.5

0.7

1

of 10

What is cos(90Ā°)?

0

0.5

1

-1

of 10

What is the value of sin(90Ā°)?

0

0.5

1

2

of 10

What is the value of sin(45Ā°)?

0.5

0.707

0.866

1

of 10

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