text: # If tan A = 5 and tan B = 4, then find the value of tan(A – B) and tan(A + B)
  
  • Last Updated : 20 Oct, 2021
  
  Trigonometry is a discipline of mathematics that studies the relationships between the lengths of the sides and angles of a right-angled triangle. Trigonometric functions, also known as goniometric functions, angle functions, or circular functions, are functions that establish the relationship between an angle to the ratio of two of the sides of a right-angled triangle. The six main trigonometric functions are sine, cosine, tangent, cotangent, secant, or cosecant.
  
  Angles defined by the ratios of trigonometric functions are known as trigonometry angles. Trigonometric angles represent trigonometric functions. The value of the angle can be anywhere between 0-360°.
  
  Attention reader! All those who say programming isn't for kids, just haven't met the right mentors yet. Join the  Demo Class for First Step to Coding Coursespecifically designed for students of class 8 to 12.
  
  The students will get to learn more about the world of programming in these free classes which will definitely help them in making a wise career choice in the future.
  
  As given in the above figure in a right-angled triangle:
  
  • Hypotenuse: The side opposite to the right angle is the hypotenuse, It is the longest side in a right-angled triangle and opposite to the 90° angle.
  • Base: The side on which angle C lies is known as the base.
  • Perpendicular: It is the side opposite to angle C in consideration.
  
  ### Trigonometric Functions
  
  Trigonometry has 6 basic trigonometric functions, they are sine, cosine, tangent, cosecant, secant, and cotangent. Now let’s look into the trigonometric functions. The six trigonometric functions are as follows,
  
  • sine: It is defined as the ratio of perpendicular and hypotenuse and It is represented as sin θ
  • cosine: It is defined as the ratio of base and hypotenuse and it is represented as cos θ
  • tangent: It is defined as the ratio of sine and cosine of an angle. Thus the definition of tangent comes out to be the ratio of perpendicular and base and is represented as tan θ
  • cosecant: It is the reciprocal of sin θ and is represented as cosec θ.
  • secant: It is the reciprocal of cos θ and is represented as sec θ.
  • cotangent: It is the reciprocal of tan θ and is represented as cot θ.
  
  According to the above image, Trigonometric Ratios are
  
  Sin θ = Perpendicular / Hypotenuse = AB/AC
  
  Cosine θ = Base / Hypotenuse = BC / AC
  
  Tangent θ = Perpendicular / Base = AB / BC
  
  Cosecant θ = Hypotenuse / Perpendicular = AC/AB
  
  Secant θ = Hypotenuse / Base = AC/BC
  
  Cotangent θ = Base / Perpendicular = BC/AB
  
  Reciprocal Identities
  
  Sin θ = 1/ Cosec θ                    OR        Cosec θ = 1/ Sin θ
  
  Cos θ = 1/ Sec θ                       OR         Sec θ = 1 / Cos θ
  
  Cot θ = 1 / Tan θ                     OR         Tan θ = 1 / Cot θ
  
  Cot θ = Cos θ / Sin θ               OR          Tan θ = Sin θ / Cos θ
  
  Tan θ.Cot θ = 1
  
  Values of Trigonometric Ratios
  
  Trigonometric Identities of Complementary and Supplementary Angles
  
  • Complementary Angles: Pair of angles whose sum is equal to 90°
  • Supplementary Angles: Pair of angles whose sum is equal to 180°
  
  Identities of Complementary angles are
  
  • sin (90° – θ) = cos θ
  • cos (90° – θ) = sin θ
  • tan (90° – θ) = cot θ
  • cot (90° – θ) = tan θ
  • sec (90° – θ) = cosec θ
  • cosec (90° – θ) = sec θ
  
  Identities of supplementary angles
  
  • sin (180° – θ) = sin θ
  • cos (180° – θ) = – cos θ
  • tan (180° – θ) = – tan θ
  • cot  (180° – θ) = – cot θ
  • sec (180° – θ) = – sec θ
  • cosec (180° – θ) = – cosec θ
  
  ### If tan A = 5 and tan B = 4, then find the value of tan(A – B) and tan(A + B)
  
  Solution:
  
  As per the trigonometric formulas
  
  tan(A + B) = (tanA + tanB) / 1- tanAtanB
  
  = (5 + 4 ) / (1 – 5*4)
  
  = – 9/19
  
  tan (A – B) = (tanA – tanB) / 1 + tanAtanB
  
  = (5-4) / (1 + 5*4)
  
  = 1 / (1 + 20)
  
  = 1/21
  
  So, the value of tan(A – B) and tan(A + B) is 1/21 and -9/19
  
  ### Similar Questions
  
  Question 1: If tan A = 4 and tan B = 2, find the value for tan(A – B).
  
  Solution:
  
  As per the formula
  
  tan(A + B) = (tanA-tanB )/(1+tanAtanB)
  
  = (4 – 2 )/(1 + 4 × 2)
  
  = 2/9
  
  Therefore,  the value of tan(A – B) is 2/19.
  
  Question 2: If sin A = 1/2, cos A = 5, sin B = 4, cos B = 3. find the value for sin(A – B)?
  
  Solution:
  
  As per the formula
  
  sin(A – B) = sinAcosB – cosAsinB
  
  = (1/2 × 3) – (5 × 4)
  
  = 3/2 – 20
  
  = (3 – 40) / 2
  
  = -37/2
  
  So, the value for sin(A – B) is -37/2
  
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explanation: The document provides a detailed explanation of trigonometric functions and identities, followed by a specific math question and its solution. The question asks to find the value of tan(A – B) and tan(A + B) given that tan A = 5 and tan B = 4. The solution is provided, showing the step-by-step calculation for both tan(A – B) and tan(A + B).
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distilabel_metadata.content: # If tan A = 5 and tan B = 4, then find the value of tan(A – B) and tan(A + B)
  
  • Last Updated : 20 Oct, 2021
  
  Trigonometry is a discipline of mathematics that studies the relationships between the lengths of the sides and angles of a right-angled triangle. Trigonometric functions, also known as goniometric functions, angle functions, or circular functions, are functions that establish the relationship between an angle to the ratio of two of the sides of a right-angled triangle. The six main trigonometric functions are sine, cosine, tangent, cotangent, secant, or cosecant.
  
  Angles defined by the ratios of trigonometric functions are known as trigonometry angles. Trigonometric angles represent trigonometric functions. The value of the angle can be anywhere between 0-360°.
  
  Attention reader! All those who say programming isn't for kids, just haven't met the right mentors yet. Join the  Demo Class for First Step to Coding Coursespecifically designed for students of class 8 to 12.
  
  The students will get to learn more about the world of programming in these free classes which will definitely help them in making a wise career choice in the future.
  
  As given in the above figure in a right-angled triangle:
  
  • Hypotenuse: The side opposite to the right angle is the hypotenuse, It is the longest side in a right-angled triangle and opposite to the 90° angle.
  • Base: The side on which angle C lies is known as the base.
  • Perpendicular: It is the side opposite to angle C in consideration.
  
  ### Trigonometric Functions
  
  Trigonometry has 6 basic trigonometric functions, they are sine, cosine, tangent, cosecant, secant, and cotangent. Now let’s look into the trigonometric functions. The six trigonometric functions are as follows,
  
  • sine: It is defined as the ratio of perpendicular and hypotenuse and It is represented as sin θ
  • cosine: It is defined as the ratio of base and hypotenuse and it is represented as cos θ
  • tangent: It is defined as the ratio of sine and cosine of an angle. Thus the definition of tangent comes out to be the ratio of perpendicular and base and is represented as tan θ
  • cosecant: It is the reciprocal of sin θ and is represented as cosec θ.
  • secant: It is the reciprocal of cos θ and is represented as sec θ.
  • cotangent: It is the reciprocal of tan θ and is represented as cot θ.
  
  According to the above image, Trigonometric Ratios are
  
  Sin θ = Perpendicular / Hypotenuse = AB/AC
  
  Cosine θ = Base / Hypotenuse = BC / AC
  
  Tangent θ = Perpendicular / Base = AB / BC
  
  Cosecant θ = Hypotenuse / Perpendicular = AC/AB
  
  Secant θ = Hypotenuse / Base = AC/BC
  
  Cotangent θ = Base / Perpendicular = BC/AB
  
  Reciprocal Identities
  
  Sin θ = 1/ Cosec θ                    OR        Cosec θ = 1/ Sin θ
  
  Cos θ = 1/ Sec θ                       OR         Sec θ = 1 / Cos θ
  
  Cot θ = 1 / Tan θ                     OR         Tan θ = 1 / Cot θ
  
  Cot θ = Cos θ / Sin θ               OR          Tan θ = Sin θ / Cos θ
  
  Tan θ.Cot θ = 1
  
  Values of Trigonometric Ratios
  
  Trigonometric Identities of Complementary and Supplementary Angles
  
  • Complementary Angles: Pair of angles whose sum is equal to 90°
  • Supplementary Angles: Pair of angles whose sum is equal to 180°
  
  Identities of Complementary angles are
  
  • sin (90° – θ) = cos θ
  • cos (90° – θ) = sin θ
  • tan (90° – θ) = cot θ
  • cot (90° – θ) = tan θ
  • sec (90° – θ) = cosec θ
  • cosec (90° – θ) = sec θ
  
  Identities of supplementary angles
  
  • sin (180° – θ) = sin θ
  • cos (180° – θ) = – cos θ
  • tan (180° – θ) = – tan θ
  • cot  (180° – θ) = – cot θ
  • sec (180° – θ) = – sec θ
  • cosec (180° – θ) = – cosec θ
  
  ### If tan A = 5 and tan B = 4, then find the value of tan(A – B) and tan(A + B)
  
  Solution:
  
  As per the trigonometric formulas
  
  tan(A + B) = (tanA + tanB) / 1- tanAtanB
  
  = (5 + 4 ) / (1 – 5*4)
  
  = – 9/19
  
  tan (A – B) = (tanA – tanB) / 1 + tanAtanB
  
  = (5-4) / (1 + 5*4)
  
  = 1 / (1 + 20)
  
  = 1/21
  
  So, the value of tan(A – B) and tan(A + B) is 1/21 and -9/19
  
  ### Similar Questions
  
  Question 1: If tan A = 4 and tan B = 2, find the value for tan(A – B).
  
  Solution:
  
  As per the formula
  
  tan(A + B) = (tanA-tanB )/(1+tanAtanB)
  
  = (4 – 2 )/(1 + 4 × 2)
  
  = 2/9
  
  Therefore,  the value of tan(A – B) is 2/19.
  
  Question 2: If sin A = 1/2, cos A = 5, sin B = 4, cos B = 3. find the value for sin(A – B)?
  
  Solution:
  
  As per the formula
  
  sin(A – B) = sinAcosB – cosAsinB
  
  = (1/2 × 3) – (5 × 4)
  
  = 3/2 – 20
  
  = (3 – 40) / 2
  
  = -37/2
  
  So, the value for sin(A – B) is -37/2
  
  My Personal Notes arrow_drop_up
  Recommended Articles
  Page :
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document provides a detailed explanation of trigonometric functions and identities, followed by a specific math question and its solution. The question asks to find the value of tan(A – B) and tan(A + B) given that tan A = 5 and tan B = 4. The solution is provided, showing the step-by-step calculation for both tan(A – B) and tan(A + B).
  
  Label: question-and-answer