Trigonometric Ratios Table, Formulas, Definitions, Mnemonics, Problems - GMS - Learning Simply
Students' favourite free learning app with LIVE online classes, instant doubt resolution, unlimited practice for classes 6-12, personalized study app for Maths, Science, Social Studies, video e-learning, online tutorial, and more. Join Telegram

Trigonometric Ratios Table, Formulas, Definitions, Mnemonics, Problems

Trigonometry definition: Branch of Mathematics which deals with the measurement of Sides and angles of a triangle and the problems based on them..
Please wait 0 seconds...
Scroll Down and click on Go to Link for destination
Congrats! Link is Generated

Six trigonometric ratios for right angle triangle are Sine(sin), Cosecant(Cos), Tangent(Tan), Cosecant(Cos), Secant(Sec), Cotangent(Cot) respectively. We will learn the sin, cos, tan formulas for these trigonometric ratios and easy ways to memorize it.


Trigonometric Ratios Table, Formulas, Definitions, Mnemonics, Problems
Trigonometric Ratios Table, Formulas, Definitions, Mnemonics, Problems


Trigonometry definition: Branch of Mathematics which deals with the measurement of Sides and angles of a triangle and the problems based on them..

Trigonometric Ratios

The ratios of the sides of the sides of a right triangle with respect to its acute angles.

Let us take a right triangle APM as shown in Figure. Here, ∠PAM (or, in brief, angle A) is an acute angle. Note the position of the side PM with respect to angle A. It faces ∠ A. We call it the side opposite to angle A. AP is the hypotenuse of the right triangle and the side AM is a part of ∠ A. So, we call it the side adjacent to angle A.
Right angled Triangle Trigonometric Ratios

∠ A = θ, AP = r (Hypotenuse) and PM = y (Perpendicular), AM = x (Base), ∠ PMA = 90o

Angle: A figure generated by rotating a given ray along of its end point.

Measurement of an Angle: Amount of rotation of the ray from initial position to the terminal position.


Trigonometric Ratios The ratios of the sides of the sides of a right triangle with respect to its acute angles.  Let us take a right triangle APM as shown in Figure. Here, ∠PAM (or, in brief, angle A) is an acute angle. Note the position of the side PM with respect to angle A. It faces ∠ A. We call it the side opposite to angle A. AP is the hypotenuse of the right triangle and the side AM is a part of ∠ A. So, we call it the side adjacent to angle A. Right angled Triangle Trigonometric Ratios  ∠ A = θ, AP = r (Hypotenuse) and PM = y (Perpendicular), AM = x (Base), ∠ PMA = 90o  Angle: A figure generated by rotating a given ray along of its end point.  Measurement of an Angle: Amount of rotation of the ray from initial position to the terminal position.  Hypotenuse Definition: the longest side of a right-angled triangle, opposite the right angle.  Perpendicular: at an angle of 90° to a given line, plane, or surface or to the ground.  Base: Side on which right angle triangle stands is known as its base  The Trigonometry Ratios of the angle θ in the triangle APM are defined as follows. Trigonometric Ratios  Opposite over Hypotenuse – Sin, Adjacent Over Hypotenuse – Cos, Opposite over Adjacent – Tan, Hypotenuse over Opposite – Cosec, Hypotenuse Over Adjacent – Sec and Adjacent over Opposite – Cotangent,  The ratios defined above are abbreviated as sin θ, cos θ, tan θ, cosec θ, sec θ and cot θ respectively. Note that the ratios cosec θ, sec θ and cot θ are respectively, the reciprocals of the ratios sin θ, cos θ and tan θ. So, the trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the length of its sides.  Opposite of Sin: Cosecant  Opposite of Cos: Secant  Opposite of Tan: Cotangent  Opposite of Cosecant: Sin  Opposite of Cotangent:  Tan  Opposite of Secant: Cosecant  Trig Mnemonics –  Some People Have, Curly Black Hair Through Proper Brushing.  Here, Some People Have is for  Sin θ= Perpendicular/ Hypotenuse. Curly Black Hair is for  Cos θ= Base/ Hypotenuse. Through Proper Brushing is for  Tan θ= Perpendicular/Base Trigonometric Ratios of Some Specific Angles We already know about isosceles right angle triangle and right angle triangle with angles 30º, 60º and 90º. Can we find sin 30º or tan 60º or cos 45º etc. with the help of these triangles? Does sin 0º or cos 0º exist?  Trigonometric Ratios of Angles Sin Cos Tan Chart Trig Table Trig Ratios of Complementary Angles We know complementary angles are pair of angles whose sum is 90° Like 40°and 50°; 60°and 30°; 20°and 70°; 15° and 75° ; etc.  sin (90° – θ) = cos θ cot (90° – θ) = tanθ cos (90° – θ) = sin θ sec (90° – θ) = cosec θ tan (90° – θ) = cot θ cosec (90° – θ) = sec θ Trigonometric Ratios Complementary Angles Table Trig Ratios Complementary Angles  Origin of Trigonometric Ratios The first use of the idea of ‘sine’ in the way we use it today was in the work Aryabhatiyam by Aryabhata, in A.D. 500. Aryabhata used the word ardha-jya for the half-chord, which was shortened to jya or jiva in due course. When the Aryabhatiyam was translated into Arabic, the word jiva was retained as it is. The word jiva was translated into sinus, which means curve, when the Arabic version was translated into Latin. Soon the word sinus, also used as sine, became common in mathematical texts throughout Europe. An English Professor of astronomy Edmund Gunter (1581–1626), first used the abbreviated notation ‘sin’.  The origin of the terms ‘cosine’ and ‘tangent’ was much later. The cosine function arose from the need to compute the sine of the complementary angle. Aryabhatta called it kotijya. The name cosinus originated with Edmund Gunter. In 1674, the English Mathematician Sir Jonas Moore first used the abbreviated notation ‘cos’.  Sin Cos Tan are the main functions used in Trigonometry and are based on a Right-Angled Triangle.  Solved Examples on Trig Ratios: Example-1. If tan A = 3/4 , then find the other trigonometric ratio of angle A. Solution : Trigonometry Ratios Questions Given tan A = 3/4 Hence tan A = Opposite side/Adjacent side = 3/4 Therefore, opposite side : adjacent side = 3:4 For angle A, opposite side = BC = 3k Adjacent side = AB = 4k (where k is any positive number) Now, we have in triangle ABC (by Pythagoras theorem) Trig Ratio Example  Example 2: If ∠ A and ∠ P are acute angles such that sin A = sin P then prove that ∠ A = ∠ P Solution : Given sin A = sin P Trig Ratios 1 Trig Ratios 2 Trig Ratios 3 Example 3: In ∆ABC, right angle is at B, AB = 5 cm and ∠ACB = 30o. Determine the lengths of the sides BC and AC. Solution: Given AB=5 cm and ∠ACB=30o.  To find the length of side BC, we will choose the trignometric ratio involving BC and the given side AB. Since BC is the side adjacent to angle C and AB is the side opposite to angle C. Therefore, AB/BC = tan C Trig Table Problem 1Trig Table Problem 2  Example 4: A chord of a circle of radius 6cm is making an angle 60o at the centre. Find the length of the chord. Solution: Given the radius of the circle OA = OB = 6cm ∠ AOB = 60o Trig Table Problem 3  Example-5. In ∆PQR, right angle is at Q, PQ = 3 cm and PR = 6 cm. Determine ∠QPR and ∠PRQ. Solution : Given PQ = 3 cm and PR = 6 cm Trigonometry Table Problems 4 Trigonometry Table Problems 5 Note : If one of the sides and any other part (either an acute angle or any side) of a right angle triangle is known, the remaining sides and angles of the triangle can be determined.  You can easily remember all trigonometry formulas using super magical hexagon, great way to remember all formulas easily.

Hypotenuse Definition: the longest side of a right-angled triangle, opposite the right angle.

Perpendicular: at an angle of 90° to a given line, plane, or surface or to the ground.

Base: Side on which right angle triangle stands is known as its base

The Trigonometry Ratios of the angle θ in the triangle APM are defined as follows.
Trigonometric Ratios

Opposite over Hypotenuse – Sin, Adjacent Over Hypotenuse – Cos, Opposite over Adjacent – Tan, Hypotenuse over Opposite – Cosec, Hypotenuse Over Adjacent – Sec and Adjacent over Opposite – Cotangent,

The ratios defined above are abbreviated as sin θ, cos θ, tan θ, cosec θ, sec θ and cot θ respectively. Note that the ratios cosec θ, sec θ and cot θ are respectively, the reciprocals of the ratios sin θ, cos θ and tan θ. So, the trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the length of its sides.

Trigonometric Ratios The ratios of the sides of the sides of a right triangle with respect to its acute angles.  Let us take a right triangle APM as shown in Figure. Here, ∠PAM (or, in brief, angle A) is an acute angle. Note the position of the side PM with respect to angle A. It faces ∠ A. We call it the side opposite to angle A. AP is the hypotenuse of the right triangle and the side AM is a part of ∠ A. So, we call it the side adjacent to angle A. Right angled Triangle Trigonometric Ratios  ∠ A = θ, AP = r (Hypotenuse) and PM = y (Perpendicular), AM = x (Base), ∠ PMA = 90o  Angle: A figure generated by rotating a given ray along of its end point.  Measurement of an Angle: Amount of rotation of the ray from initial position to the terminal position.  Hypotenuse Definition: the longest side of a right-angled triangle, opposite the right angle.  Perpendicular: at an angle of 90° to a given line, plane, or surface or to the ground.  Base: Side on which right angle triangle stands is known as its base  The Trigonometry Ratios of the angle θ in the triangle APM are defined as follows. Trigonometric Ratios  Opposite over Hypotenuse – Sin, Adjacent Over Hypotenuse – Cos, Opposite over Adjacent – Tan, Hypotenuse over Opposite – Cosec, Hypotenuse Over Adjacent – Sec and Adjacent over Opposite – Cotangent,  The ratios defined above are abbreviated as sin θ, cos θ, tan θ, cosec θ, sec θ and cot θ respectively. Note that the ratios cosec θ, sec θ and cot θ are respectively, the reciprocals of the ratios sin θ, cos θ and tan θ. So, the trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the length of its sides.  Opposite of Sin: Cosecant  Opposite of Cos: Secant  Opposite of Tan: Cotangent  Opposite of Cosecant: Sin  Opposite of Cotangent:  Tan  Opposite of Secant: Cosecant  Trig Mnemonics –  Some People Have, Curly Black Hair Through Proper Brushing.  Here, Some People Have is for  Sin θ= Perpendicular/ Hypotenuse. Curly Black Hair is for  Cos θ= Base/ Hypotenuse. Through Proper Brushing is for  Tan θ= Perpendicular/Base Trigonometric Ratios of Some Specific Angles We already know about isosceles right angle triangle and right angle triangle with angles 30º, 60º and 90º. Can we find sin 30º or tan 60º or cos 45º etc. with the help of these triangles? Does sin 0º or cos 0º exist?  Trigonometric Ratios of Angles Sin Cos Tan Chart Trig Table Trig Ratios of Complementary Angles We know complementary angles are pair of angles whose sum is 90° Like 40°and 50°; 60°and 30°; 20°and 70°; 15° and 75° ; etc.  sin (90° – θ) = cos θ cot (90° – θ) = tanθ cos (90° – θ) = sin θ sec (90° – θ) = cosec θ tan (90° – θ) = cot θ cosec (90° – θ) = sec θ Trigonometric Ratios Complementary Angles Table Trig Ratios Complementary Angles  Origin of Trigonometric Ratios The first use of the idea of ‘sine’ in the way we use it today was in the work Aryabhatiyam by Aryabhata, in A.D. 500. Aryabhata used the word ardha-jya for the half-chord, which was shortened to jya or jiva in due course. When the Aryabhatiyam was translated into Arabic, the word jiva was retained as it is. The word jiva was translated into sinus, which means curve, when the Arabic version was translated into Latin. Soon the word sinus, also used as sine, became common in mathematical texts throughout Europe. An English Professor of astronomy Edmund Gunter (1581–1626), first used the abbreviated notation ‘sin’.  The origin of the terms ‘cosine’ and ‘tangent’ was much later. The cosine function arose from the need to compute the sine of the complementary angle. Aryabhatta called it kotijya. The name cosinus originated with Edmund Gunter. In 1674, the English Mathematician Sir Jonas Moore first used the abbreviated notation ‘cos’.  Sin Cos Tan are the main functions used in Trigonometry and are based on a Right-Angled Triangle.  Solved Examples on Trig Ratios: Example-1. If tan A = 3/4 , then find the other trigonometric ratio of angle A. Solution : Trigonometry Ratios Questions Given tan A = 3/4 Hence tan A = Opposite side/Adjacent side = 3/4 Therefore, opposite side : adjacent side = 3:4 For angle A, opposite side = BC = 3k Adjacent side = AB = 4k (where k is any positive number) Now, we have in triangle ABC (by Pythagoras theorem) Trig Ratio Example  Example 2: If ∠ A and ∠ P are acute angles such that sin A = sin P then prove that ∠ A = ∠ P Solution : Given sin A = sin P Trig Ratios 1 Trig Ratios 2 Trig Ratios 3 Example 3: In ∆ABC, right angle is at B, AB = 5 cm and ∠ACB = 30o. Determine the lengths of the sides BC and AC. Solution: Given AB=5 cm and ∠ACB=30o.  To find the length of side BC, we will choose the trignometric ratio involving BC and the given side AB. Since BC is the side adjacent to angle C and AB is the side opposite to angle C. Therefore, AB/BC = tan C Trig Table Problem 1Trig Table Problem 2  Example 4: A chord of a circle of radius 6cm is making an angle 60o at the centre. Find the length of the chord. Solution: Given the radius of the circle OA = OB = 6cm ∠ AOB = 60o Trig Table Problem 3  Example-5. In ∆PQR, right angle is at Q, PQ = 3 cm and PR = 6 cm. Determine ∠QPR and ∠PRQ. Solution : Given PQ = 3 cm and PR = 6 cm Trigonometry Table Problems 4 Trigonometry Table Problems 5 Note : If one of the sides and any other part (either an acute angle or any side) of a right angle triangle is known, the remaining sides and angles of the triangle can be determined.  You can easily remember all trigonometry formulas using super magical hexagon, great way to remember all formulas easily.

Opposite of Sin: Cosecant

Opposite of Cos: Secant

Opposite of Tan: Cotangent

Opposite of Cosecant: Sin

Opposite of Cotangent:  Tan

Opposite of Secant: Cosecant

Trig Mnemonics –  Some People Have, Curly Black Hair Through Proper Brushing.

Here, Some People Have is for

  • Sin θ= Perpendicular/ Hypotenuse.

Curly Black Hair is for

  • Cos θ= Base/ Hypotenuse.

Through Proper Brushing is for

  • Tan θ= Perpendicular/Base

Trigonometric Ratios of Some Specific Angles

We already know about isosceles right angle triangle and right angle triangle with angles 30º, 60º and 90º.
Can we find sin 30º or tan 60º or cos 45º etc. with the help of these triangles?
Does sin 0º or cos 0º exist?

Trigonometric Ratios of Angles Sin Cos Tan Chart
Trig Table

Trigonometric Ratios The ratios of the sides of the sides of a right triangle with respect to its acute angles.  Let us take a right triangle APM as shown in Figure. Here, ∠PAM (or, in brief, angle A) is an acute angle. Note the position of the side PM with respect to angle A. It faces ∠ A. We call it the side opposite to angle A. AP is the hypotenuse of the right triangle and the side AM is a part of ∠ A. So, we call it the side adjacent to angle A. Right angled Triangle Trigonometric Ratios  ∠ A = θ, AP = r (Hypotenuse) and PM = y (Perpendicular), AM = x (Base), ∠ PMA = 90o  Angle: A figure generated by rotating a given ray along of its end point.  Measurement of an Angle: Amount of rotation of the ray from initial position to the terminal position.  Hypotenuse Definition: the longest side of a right-angled triangle, opposite the right angle.  Perpendicular: at an angle of 90° to a given line, plane, or surface or to the ground.  Base: Side on which right angle triangle stands is known as its base  The Trigonometry Ratios of the angle θ in the triangle APM are defined as follows. Trigonometric Ratios  Opposite over Hypotenuse – Sin, Adjacent Over Hypotenuse – Cos, Opposite over Adjacent – Tan, Hypotenuse over Opposite – Cosec, Hypotenuse Over Adjacent – Sec and Adjacent over Opposite – Cotangent,  The ratios defined above are abbreviated as sin θ, cos θ, tan θ, cosec θ, sec θ and cot θ respectively. Note that the ratios cosec θ, sec θ and cot θ are respectively, the reciprocals of the ratios sin θ, cos θ and tan θ. So, the trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the length of its sides.  Opposite of Sin: Cosecant  Opposite of Cos: Secant  Opposite of Tan: Cotangent  Opposite of Cosecant: Sin  Opposite of Cotangent:  Tan  Opposite of Secant: Cosecant  Trig Mnemonics –  Some People Have, Curly Black Hair Through Proper Brushing.  Here, Some People Have is for  Sin θ= Perpendicular/ Hypotenuse. Curly Black Hair is for  Cos θ= Base/ Hypotenuse. Through Proper Brushing is for  Tan θ= Perpendicular/Base Trigonometric Ratios of Some Specific Angles We already know about isosceles right angle triangle and right angle triangle with angles 30º, 60º and 90º. Can we find sin 30º or tan 60º or cos 45º etc. with the help of these triangles? Does sin 0º or cos 0º exist?  Trigonometric Ratios of Angles Sin Cos Tan Chart Trig Table Trig Ratios of Complementary Angles We know complementary angles are pair of angles whose sum is 90° Like 40°and 50°; 60°and 30°; 20°and 70°; 15° and 75° ; etc.  sin (90° – θ) = cos θ cot (90° – θ) = tanθ cos (90° – θ) = sin θ sec (90° – θ) = cosec θ tan (90° – θ) = cot θ cosec (90° – θ) = sec θ Trigonometric Ratios Complementary Angles Table Trig Ratios Complementary Angles  Origin of Trigonometric Ratios The first use of the idea of ‘sine’ in the way we use it today was in the work Aryabhatiyam by Aryabhata, in A.D. 500. Aryabhata used the word ardha-jya for the half-chord, which was shortened to jya or jiva in due course. When the Aryabhatiyam was translated into Arabic, the word jiva was retained as it is. The word jiva was translated into sinus, which means curve, when the Arabic version was translated into Latin. Soon the word sinus, also used as sine, became common in mathematical texts throughout Europe. An English Professor of astronomy Edmund Gunter (1581–1626), first used the abbreviated notation ‘sin’.  The origin of the terms ‘cosine’ and ‘tangent’ was much later. The cosine function arose from the need to compute the sine of the complementary angle. Aryabhatta called it kotijya. The name cosinus originated with Edmund Gunter. In 1674, the English Mathematician Sir Jonas Moore first used the abbreviated notation ‘cos’.  Sin Cos Tan are the main functions used in Trigonometry and are based on a Right-Angled Triangle.  Solved Examples on Trig Ratios: Example-1. If tan A = 3/4 , then find the other trigonometric ratio of angle A. Solution : Trigonometry Ratios Questions Given tan A = 3/4 Hence tan A = Opposite side/Adjacent side = 3/4 Therefore, opposite side : adjacent side = 3:4 For angle A, opposite side = BC = 3k Adjacent side = AB = 4k (where k is any positive number) Now, we have in triangle ABC (by Pythagoras theorem) Trig Ratio Example  Example 2: If ∠ A and ∠ P are acute angles such that sin A = sin P then prove that ∠ A = ∠ P Solution : Given sin A = sin P Trig Ratios 1 Trig Ratios 2 Trig Ratios 3 Example 3: In ∆ABC, right angle is at B, AB = 5 cm and ∠ACB = 30o. Determine the lengths of the sides BC and AC. Solution: Given AB=5 cm and ∠ACB=30o.  To find the length of side BC, we will choose the trignometric ratio involving BC and the given side AB. Since BC is the side adjacent to angle C and AB is the side opposite to angle C. Therefore, AB/BC = tan C Trig Table Problem 1Trig Table Problem 2  Example 4: A chord of a circle of radius 6cm is making an angle 60o at the centre. Find the length of the chord. Solution: Given the radius of the circle OA = OB = 6cm ∠ AOB = 60o Trig Table Problem 3  Example-5. In ∆PQR, right angle is at Q, PQ = 3 cm and PR = 6 cm. Determine ∠QPR and ∠PRQ. Solution : Given PQ = 3 cm and PR = 6 cm Trigonometry Table Problems 4 Trigonometry Table Problems 5 Note : If one of the sides and any other part (either an acute angle or any side) of a right angle triangle is known, the remaining sides and angles of the triangle can be determined.  You can easily remember all trigonometry formulas using super magical hexagon, great way to remember all formulas easily.

Trig Ratios of Complementary Angles

We know complementary angles are pair of angles whose sum is 90° Like 40°and 50°; 60°and 30°; 20°and 70°; 15° and 75° ; etc.

  • sin (90° – θ) = cos θ
  • cot (90° – θ) = tanθ
  • cos (90° – θ) = sin θ
  • sec (90° – θ) = cosec θ
  • tan (90° – θ) = cot θ
  • cosec (90° – θ) = sec θ

Trigonometric Ratios Complementary Angles Table Trig Ratios Complementary Angles

Origin of Trigonometric Ratios

The first use of the idea of ‘sine’ in the way we use it today was in the work Aryabhatiyam by Aryabhata, in A.D. 500. Aryabhata used the word ardha-jya for the half-chord, which was shortened to jya or jiva in due course. When the Aryabhatiyam was translated into Arabic, the word jiva was retained as it is. The word jiva was translated into sinus, which means curve, when the Arabic version was translated into Latin. Soon the word sinus, also used as sine,
became common in mathematical texts throughout Europe. An English Professor of astronomy Edmund Gunter (1581–1626), first used the abbreviated notation ‘sin’.

The origin of the terms ‘cosine’ and ‘tangent’ was much later. The cosine function arose from the need to compute the sine of the complementary angle. Aryabhatta called it kotijya. The name cosinus originated with Edmund Gunter. In 1674, the English Mathematician Sir Jonas Moore first used the abbreviated notation ‘cos’.

Sin Cos Tan are the main functions used in Trigonometry and are based on a Right-Angled Triangle.

Solved Examples on Trig Ratios:

Example-1If tan A = 3/4 , then find the other trigonometric ratio of angle A.
Solution :
Trigonometry Ratios Questions
Given tan A = 3/4
Hence tan A = Opposite side/Adjacent side = 3/4
Therefore, opposite side : adjacent side = 3:4
For angle A, opposite side = BC = 3k
Adjacent side = AB = 4(where is any positive number)
Now, we have in triangle ABC (by Pythagoras theorem)
Trig Ratio Example

Example 2: If ∠ A and ∠ P are acute angles such that sin A = sin P then prove that ∠ A = ∠ P

Solution : Given sin A = sin P
Trig Ratios 1
Trig Ratios 2
Trig Ratios 3
Example 3: In ∆ABC, right angle is at B, AB = 5 cm and ∠ACB = 30o. Determine the lengths of the sides BC and AC.
Solution: Given AB=5 cm and
∠ACB=30o.

To find the length of side BC, we will choose the trignometric ratio involving BC and the given side AB. Since BC is the side adjacent to angle C and AB is the side opposite to angle C.
Therefore,
AB/BC = tan C
Trig Table Problem 1Trig Table Problem 2

Example 4: A chord of a circle of radius 6cm is making an angle 60o at the centre. Find the length of the chord.
Solution: Given the radius of the circle OA = OB = 6cm
∠ AOB = 60o
Trig Table Problem 3

Example-5. In ∆PQR, right angle is at Q, PQ = 3 cm and PR = 6 cm. Determine ∠QPR and ∠PRQ.
Solution : Given PQ = 3 cm and PR = 6 cm
Trigonometry Table Problems 4
Trigonometry Table Problems 5
Note : If one of the sides and any other part (either an acute angle or any side) of a right angle triangle is known, the remaining sides and angles of the triangle can be determined.

You can easily remember all trigonometry formulas using super magical hexagon, great way to remember all formulas easily.

Trigonometric Ratios The ratios of the sides of the sides of a right triangle with respect to its acute angles.  Let us take a right triangle APM as shown in Figure. Here, ∠PAM (or, in brief, angle A) is an acute angle. Note the position of the side PM with respect to angle A. It faces ∠ A. We call it the side opposite to angle A. AP is the hypotenuse of the right triangle and the side AM is a part of ∠ A. So, we call it the side adjacent to angle A. Right angled Triangle Trigonometric Ratios  ∠ A = θ, AP = r (Hypotenuse) and PM = y (Perpendicular), AM = x (Base), ∠ PMA = 90o  Angle: A figure generated by rotating a given ray along of its end point.  Measurement of an Angle: Amount of rotation of the ray from initial position to the terminal position.  Hypotenuse Definition: the longest side of a right-angled triangle, opposite the right angle.  Perpendicular: at an angle of 90° to a given line, plane, or surface or to the ground.  Base: Side on which right angle triangle stands is known as its base  The Trigonometry Ratios of the angle θ in the triangle APM are defined as follows. Trigonometric Ratios  Opposite over Hypotenuse – Sin, Adjacent Over Hypotenuse – Cos, Opposite over Adjacent – Tan, Hypotenuse over Opposite – Cosec, Hypotenuse Over Adjacent – Sec and Adjacent over Opposite – Cotangent,  The ratios defined above are abbreviated as sin θ, cos θ, tan θ, cosec θ, sec θ and cot θ respectively. Note that the ratios cosec θ, sec θ and cot θ are respectively, the reciprocals of the ratios sin θ, cos θ and tan θ. So, the trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the length of its sides.  Opposite of Sin: Cosecant  Opposite of Cos: Secant  Opposite of Tan: Cotangent  Opposite of Cosecant: Sin  Opposite of Cotangent:  Tan  Opposite of Secant: Cosecant  Trig Mnemonics –  Some People Have, Curly Black Hair Through Proper Brushing.  Here, Some People Have is for  Sin θ= Perpendicular/ Hypotenuse. Curly Black Hair is for  Cos θ= Base/ Hypotenuse. Through Proper Brushing is for  Tan θ= Perpendicular/Base Trigonometric Ratios of Some Specific Angles We already know about isosceles right angle triangle and right angle triangle with angles 30º, 60º and 90º. Can we find sin 30º or tan 60º or cos 45º etc. with the help of these triangles? Does sin 0º or cos 0º exist?  Trigonometric Ratios of Angles Sin Cos Tan Chart Trig Table Trig Ratios of Complementary Angles We know complementary angles are pair of angles whose sum is 90° Like 40°and 50°; 60°and 30°; 20°and 70°; 15° and 75° ; etc.  sin (90° – θ) = cos θ cot (90° – θ) = tanθ cos (90° – θ) = sin θ sec (90° – θ) = cosec θ tan (90° – θ) = cot θ cosec (90° – θ) = sec θ Trigonometric Ratios Complementary Angles Table Trig Ratios Complementary Angles  Origin of Trigonometric Ratios The first use of the idea of ‘sine’ in the way we use it today was in the work Aryabhatiyam by Aryabhata, in A.D. 500. Aryabhata used the word ardha-jya for the half-chord, which was shortened to jya or jiva in due course. When the Aryabhatiyam was translated into Arabic, the word jiva was retained as it is. The word jiva was translated into sinus, which means curve, when the Arabic version was translated into Latin. Soon the word sinus, also used as sine, became common in mathematical texts throughout Europe. An English Professor of astronomy Edmund Gunter (1581–1626), first used the abbreviated notation ‘sin’.  The origin of the terms ‘cosine’ and ‘tangent’ was much later. The cosine function arose from the need to compute the sine of the complementary angle. Aryabhatta called it kotijya. The name cosinus originated with Edmund Gunter. In 1674, the English Mathematician Sir Jonas Moore first used the abbreviated notation ‘cos’.  Sin Cos Tan are the main functions used in Trigonometry and are based on a Right-Angled Triangle.  Solved Examples on Trig Ratios: Example-1. If tan A = 3/4 , then find the other trigonometric ratio of angle A. Solution : Trigonometry Ratios Questions Given tan A = 3/4 Hence tan A = Opposite side/Adjacent side = 3/4 Therefore, opposite side : adjacent side = 3:4 For angle A, opposite side = BC = 3k Adjacent side = AB = 4k (where k is any positive number) Now, we have in triangle ABC (by Pythagoras theorem) Trig Ratio Example  Example 2: If ∠ A and ∠ P are acute angles such that sin A = sin P then prove that ∠ A = ∠ P Solution : Given sin A = sin P Trig Ratios 1 Trig Ratios 2 Trig Ratios 3 Example 3: In ∆ABC, right angle is at B, AB = 5 cm and ∠ACB = 30o. Determine the lengths of the sides BC and AC. Solution: Given AB=5 cm and ∠ACB=30o.  To find the length of side BC, we will choose the trignometric ratio involving BC and the given side AB. Since BC is the side adjacent to angle C and AB is the side opposite to angle C. Therefore, AB/BC = tan C Trig Table Problem 1Trig Table Problem 2  Example 4: A chord of a circle of radius 6cm is making an angle 60o at the centre. Find the length of the chord. Solution: Given the radius of the circle OA = OB = 6cm ∠ AOB = 60o Trig Table Problem 3  Example-5. In ∆PQR, right angle is at Q, PQ = 3 cm and PR = 6 cm. Determine ∠QPR and ∠PRQ. Solution : Given PQ = 3 cm and PR = 6 cm Trigonometry Table Problems 4 Trigonometry Table Problems 5 Note : If one of the sides and any other part (either an acute angle or any side) of a right angle triangle is known, the remaining sides and angles of the triangle can be determined.  You can easily remember all trigonometry formulas using super magical hexagon, great way to remember all formulas easily.

About the Author

At the helm of GMS Learning is Principal Balkishan Agrawal, a dedicated and experienced educationist. Under his able guidance, our school has flourished academically and has achieved remarkable milestones in various fields. Principal Agrawal’s visio…

Post a Comment

Cookie Consent
We serve cookies on this site to analyze traffic, remember your preferences, and optimize your experience.
Oops!
It seems there is something wrong with your internet connection. Please connect to the internet and start browsing again.
AdBlock Detected!
We have detected that you are using adblocking plugin in your browser.
The revenue we earn by the advertisements is used to manage this website, we request you to whitelist our website in your adblocking plugin.
Site is Blocked
Sorry! This site is not available in your country.