- Today’s Topic: Addition formula of tan inverse function
- Objective: To derive the addition formula for tan inverse function
- Understanding the concept of inverse trigonometric functions
- Review of addition formulas in trigonometry
- Definition of tan inverse function
- Deriving the addition formula for tan inverse function:
- Consider two angles, θ1 and θ2
- Let a = tan(θ1) and b = tan(θ2)
- Hence, θ1 = tan^(-1)(a) and θ2 = tan^(-1)(b)
- Using the addition formula of tan function, we can express tan(θ1 + θ2) in terms of a and b
- Applying the inverse tan function on both sides, we get the addition formula for tan inverse function
- Examples to understand the application of addition formula
- Find the value of tan^(-1)(2) + tan^(-1)(1)
- Solve for tan^(-1)(3/4) + tan^(-1)(5/12)
- Applications of addition formula in solving trigonometric equations
- Summary of the addition formula for tan inverse function
Slide 11: Understanding the concept of inverse trigonometric functions
- Inverse trigonometric functions are the functions used to find the angle corresponding to a given trigonometric ratio.
- They are denoted by the prefix “arc” or by the abbreviation “inv” followed by the trigonometric function (e.g. arcsin, arccos, arctan).
- The output of inverse trigonometric functions is an angle, expressed in radians or degrees.
- Inverse trigonometric functions are used in various branches of mathematics and in real-life applications, such as physics, engineering, and computer science.
- The key inverse trigonometric functions are arcsin (sin^(-1)), arccos (cos^(-1)), and arctan (tan^(-1)).
- Addition formulas in trigonometry are used to find the sum of trigonometric functions of two angles.
- The most commonly used addition formulas include:
- Sinusoidal Addition Formulas:
- sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
- cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
- Tangent Addition Formula:
- tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
- Addition formulas are used to simplify trigonometric expressions, solve trigonometric equations, and establish relationships between different trigonometric functions.
Slide 13: Definition of tan inverse function
- The tan inverse function, denoted as tan^(-1) or arctan, is the inverse of the tangent function.
- It is used to find the angle whose tangent is a given value.
- The tan inverse function takes a ratio as input and returns an angle as output.
- The range of tan inverse function is (-π/2, π/2), which means it returns angles between -90 degrees and 90 degrees.
- The domain of tan inverse function is all real numbers.
- To derive the addition formula for tan inverse function, we start with two angles θ1 and θ2.
- Let a = tan(θ1) and b = tan(θ2).
- Therefore, θ1 = tan^(-1)(a) and θ2 = tan^(-1)(b).
- Using the addition formula for tangent function, tan(θ1 + θ2), we can express it in terms of a and b.
- Applying the inverse tangent function (tan^(-1)) on both sides, we obtain the addition formula for tan inverse function.
- Deriving the addition formula involves the following steps:
- Start with the sum of two angles: θ1 + θ2
- Use the tangent addition formula: tan(θ1 + θ2) = (tan(θ1) + tan(θ2)) / (1 - tan(θ1)tan(θ2))
- Replace tan(θ1) with a and tan(θ2) with b
- Apply the inverse tangent function (tan^(-1)) on both sides to get the addition formula for tan inverse function.
- Example 1: Find the value of tan^(-1)(2) + tan^(-1)(1)
- Example 2: Solve for tan^(-1)(3/4) + tan^(-1)(5/12)
- By applying the addition formula for tan inverse function, we can find the exact value or simplify the expression involving tan inverse functions.
- These examples help us practice the addition formula and familiarize ourselves with its application.
Slide 17: Example 1: Find the value of tan^(-1)(2) + tan^(-1)(1)
- Given: tan^(-1)(2) + tan^(-1)(1)
- Using the addition formula, we substitute a = 2 and b = 1:
- tan^(-1)(2) + tan^(-1)(1) = tan^(-1)((2 + 1) / (1 - 2 * 1))
- tan^(-1)(2) + tan^(-1)(1) = tan^(-1)(3 / (-1))
- tan^(-1)(2) + tan^(-1)(1) = tan^(-1)(-3)
- Therefore, tan^(-1)(2) + tan^(-1)(1) = -tan^(-1)(3)
Slide 18: Example 2: Solve for tan^(-1)(3/4) + tan^(-1)(5/12)
- Given: tan^(-1)(3/4) + tan^(-1)(5/12)
- Using the addition formula, we substitute a = 3/4 and b = 5/12:
- tan^(-1)(3/4) + tan^(-1)(5/12) = tan^(-1)(((3/4) + (5/12)) / (1 - (3/4) * (5/12)))
- tan^(-1)(3/4) + tan^(-1)(5/12) = tan^(-1)((9/12 + 5/12) / (1 - 15/48))
- tan^(-1)(3/4) + tan^(-1)(5/12) = tan^(-1)(14/12 / (33/48))
- tan^(-1)(3/4) + tan^(-1)(5/12) = tan^(-1)(14/12 * 48/33)
- Therefore, we obtain the simplified expression for tan^(-1)(3/4) + tan^(-1)(5/12) as well.
- Addition formulas for tan inverse function are used to solve trigonometric equations involving tan inverse terms.
- These equations may arise in various applications, such as physics, engineering, and finance.
- By applying the addition formula, we can simplify the equation and find the values of the unknown angles.
- The addition formula helps in establishing relationships between trigonometric functions and solving complex trigonometric equations.
- The addition formula for tan inverse function is derived by applying the inverse tangent function (tan^(-1)) on both sides of the tangent addition formula.
- It helps to find the sum of two angles when given the tan inverse values of those angles.
- The addition formula is helpful in simplifying trigonometric expressions involving tan inverse terms and in solving trigonometric equations.
- Examples demonstrate the application of the addition formula for tan inverse function in solving problems.
Slide 21: Properties of tan inverse function
- The tan inverse function has the following properties:
- tan^(-1)(0) = 0: The tangent of 0 degrees or radians is 0, thus tan^(-1)(0) returns 0.
- tan^(-1)(∞) = π/2: The tangent of 90 degrees or π/2 radians is undefined (∞), thus tan^(-1)(∞) returns π/2.
- tan^(-1)(-∞) = -π/2: The tangent of -90 degrees or -π/2 radians is undefined (-∞), thus tan^(-1)(-∞) returns -π/2.
- tan^(-1)(1) = π/4: The tangent of 45 degrees or π/4 radians is 1, thus tan^(-1)(1) returns π/4.
- tan^(-1)(-1) = -π/4: The tangent of -45 degrees or -π/4 radians is -1, thus tan^(-1)(-1) returns -π/4.
Slide 22: Domain and range of tan inverse function
- The domain of tan inverse function is all real numbers (-∞, ∞).
- The range of tan inverse function is (-π/2, π/2), which means it returns angles between -90 degrees and 90 degrees.
Slide 23: Domain restrictions of tan inverse function
- Tan inverse function has restrictions on its domain due to the nature of tangent.
- The tangent function is not defined for certain angles:
- tan^(-1)(x) is undefined when x is less than or equal to -∞ or greater than or equal to +∞.
- tan^(-1)(x) is undefined when x is outside the range of (-π/2, π/2).
- Thus, tan^(-1)(x) is not defined when x is less than -1 or greater than +1.
- The subtraction formula for tan inverse function can be derived in a similar way as the addition formula.
- To derive the subtraction formula, we start with two angles θ1 and θ2.
- Let a = tan(θ1) and b = tan(θ2).
- Therefore, θ1 = tan^(-1)(a) and θ2 = tan^(-1)(b).
- Using the subtraction formula for tangent function, tan(θ1 - θ2), we can express it in terms of a and b.
- Applying the inverse tangent function (tan^(-1)) on both sides, we obtain the subtraction formula for tan inverse function.
- Example 1: Find the value of tan^(-1)(2) - tan^(-1)(1)
- Example 2: Solve for tan^(-1)(3/4) - tan^(-1)(5/12)
- Similar to the addition formula, the subtraction formula for tan inverse function is used to find the exact value or simplify expressions involving tan inverse terms.
- These examples help us practice the subtraction formula and gain confidence in using it.
Slide 26: Example 1: Find the value of tan^(-1)(2) - tan^(-1)(1)
- Given: tan^(-1)(2) - tan^(-1)(1)
- Using the subtraction formula, we substitute a = 2 and b = 1:
- tan^(-1)(2) - tan^(-1)(1) = tan^(-1)((2 - 1) / (1 + 2 * 1))
- tan^(-1)(2) - tan^(-1)(1) = tan^(-1)(1 / 3)
- Therefore, tan^(-1)(2) - tan^(-1)(1) = tan^(-1)(1/3)
Slide 27: Example 2: Solve for tan^(-1)(3/4) - tan^(-1)(5/12)
- Given: tan^(-1)(3/4) - tan^(-1)(5/12)
- Using the subtraction formula, we substitute a = 3/4 and b = 5/12:
- tan^(-1)(3/4) - tan^(-1)(5/12) = tan^(-1)(((3/4) - (5/12)) / (1 + (3/4) * (5/12)))
- tan^(-1)(3/4) - tan^(-1)(5/12) = tan^(-1)((9/12 - 5/12) / (1 + 15/48))
- tan^(-1)(3/4) - tan^(-1)(5/12) = tan^(-1)(4/12 / (33/48))
- tan^(-1)(3/4) - tan^(-1)(5/12) = tan^(-1)(4/12 * 48/33)
- Therefore, we obtain the simplified expression for tan^(-1)(3/4) - tan^(-1)(5/12) as well.
- The subtraction formula for tan inverse function is derived by applying the inverse tangent function (tan^(-1)) on both sides of the tangent subtraction formula.
- It helps to find the difference between two angles when given the tan inverse values of those angles.
- The subtraction formula is useful in simplifying trigonometric expressions involving tan inverse terms and in solving trigonometric equations.
- Similar to the addition formula, examples demonstrate the application of the subtraction formula for tan inverse function.
Slide 29: Applications of tan inverse function
- Tan inverse function has various applications in mathematics and other fields, including:
- Solving right triangle problems: The tan inverse function helps find unknown angles in right triangles when the lengths of sides are given.
- Calculating slopes and angles: In physics and engineering, tan inverse function is used to calculate slopes of lines and the angles of rotation.
- Evaluating limits and derivatives: In calculus, tan inverse function is employed to find limits and derivatives of functions involving trigonometry.
- Analyzing periodic phenomena: Tan inverse function is used to analyze the behavior of periodic functions, such as harmonic oscillations.
- Interpretation of survey data: Tan inverse function aids in interpreting survey data involving angles and distances.
Slide 30: Recap and Conclusion
- Today, we covered the addition and subtraction formulas for tan inverse function.
- The addition formula allows us to find the sum of two angles given their tan inverse values.
- The subtraction formula helps us find the difference between two angles using their tan inverse values.
- These formulas simplify trigonometric expressions and assist in solving trigonometric equations.
- We also explored the properties, domain, and range of tan inverse function.
- Tan inverse function finds applications in various fields such as physics, engineering, calculus, and survey analysis.
- It is important to practice using these formulas and understand their applications to excel in trigonometry and related fields.