-
Let’s consider an equation: sin(A) + sin(B) where A and B are angles.
-
We will now derive a formula for finding the sum of the inverses of sine.
-
The formula is given by: sin^(-1)(a) + sin^(-1)(b) = sin^(-1)(ab + sqrt(1-a^2) sqrt(1-b^2))
-
The formula can be used when a, b, and the expression inside the inverse sine function are between -1 and 1.
-
Example: Find the value of sin^(-1)(3/5) + sin^(-1)(4/5).
- Using the formula, we have:
- sin^(-1)(3/5) + sin^(-1)(4/5) = sin^(-1)([3/5][4/5] + sqrt(1-(3/5)^2)sqrt(1-(4/5)^2))
- sin^(-1)(3/5) + sin^(-1)(4/5) = sin^(-1)(12/25 + sqrt(1-(9/25))(sqrt(1-(16/25))))
- sin^(-1)(3/5) + sin^(-1)(4/5) = sin^(-1)(12/25 + sqrt(16/25-9/25))
- sin^(-1)(3/5) + sin^(-1)(4/5) = sin^(-1)(12/25 + sqrt(7/25))
- sin^(-1)(3/5) + sin^(-1)(4/5) = sin^(-1)(12/25 + sqrt(7)/5)
- sin^(-1)(3/5) + sin^(-1)(4/5) = sin^(-1)((12+5sqrt(7))/25)
Inverse Trigonometric Functions - Product of inverses of sine
-
Now, let’s consider an equation: sin(A)sin(B) where A and B are angles.
-
Example: Find the value of sin^(-1)(3/5)sin^(-1)(4/5).
- Using the formula, we have:
- sin^(-1)(3/5)sin^(-1)(4/5) = sin^(-1)([3/5][4/5] - sqrt(1-(3/5)^2)sqrt(1-(4/5)^2))
- sin^(-1)(3/5)sin^(-1)(4/5) = sin^(-1)(12/25 - sqrt(1-(9/25))(sqrt(1-(16/25))))
- sin^(-1)(3/5)sin^(-1)(4/5) = sin^(-1)(12/25 - sqrt(16/25-9/25))
- sin^(-1)(3/5)sin^(-1)(4/5) = sin^(-1)(12/25 - sqrt(7/25))
- sin^(-1)(3/5)sin^(-1)(4/5) = sin^(-1)(12/25 - sqrt(7)/5)
- sin^(-1)(3/5)sin^(-1)(4/5) = sin^(-1)((12-5sqrt(7))/25)
Inverse Trigonometric Functions - Difference of inverses of sine
-
Lastly, let’s consider an equation: sin(A) - sin(B) where A and B are angles.
-
Example: Find the value of sin^(-1)(3/5) - sin^(-1)(4/5).
- Using the formula, we have:
- sin^(-1)(3/5) - sin^(-1)(4/5) = sin^(-1)([3/5][4/5] - sqrt(1-(3/5)^2)sqrt(1-(4/5)^2))
- sin^(-1)(3/5) - sin^(-1)(4/5) = sin^(-1)(12/25 - sqrt(1-(9/25))(sqrt(1-(16/25))))
- sin^(-1)(3/5) - sin^(-1)(4/5) = sin^(-1)(12/25 - sqrt(16/25-9/25))
- sin^(-1)(3/5) - sin^(-1)(4/5) = sin^(-1)(12/25 - sqrt(7/25))
- sin^(-1)(3/5) - sin^(-1)(4/5) = sin^(-1)(12/25 - sqrt(7)/5)
- sin^(-1)(3/5) - sin^(-1)(4/5) = sin^(-1)((12-5sqrt(7))/25)
In summary, we have formulas for finding the sum, product, and difference of the inverses of sine. These formulas are useful when working with angles and trigonometric functions involving sine.
Inverse Trigonometric Functions - Sum of inverses of sine
-
Let’s consider an equation: sin(A) + sin(B) where A and B are angles.
-
We will now derive a formula for finding the sum of the inverses of sine.
-
The formula is given by: sin^(-1)(a) + sin^(-1)(b) = sin^(-1)(ab + sqrt(1-a^2) sqrt(1-b^2))
-
The formula can be used when a, b, and the expression inside the inverse sine function are between -1 and 1.
-
Example: Find the value of sin^(-1)(3/5) + sin^(-1)(4/5).
- Using the formula, we have:
- sin^(-1)(3/5) + sin^(-1)(4/5) = sin^(-1)((3/5)(4/5) + sqrt(1-(3/5)^2)sqrt(1-(4/5)^2))
- sin^(-1)(3/5) + sin^(-1)(4/5) = sin^(-1)(12/25 + sqrt(1-(9/25))(sqrt(1-(16/25))))
- sin^(-1)(3/5) + sin^(-1)(4/5) = sin^(-1)(12/25 + sqrt(16/25-9/25))
- sin^(-1)(3/5) + sin^(-1)(4/5) = sin^(-1)(12/25 + sqrt(7/25))
- sin^(-1)(3/5) + sin^(-1)(4/5) = sin^(-1)(12/25 + sqrt(7)/5)
- sin^(-1)(3/5) + sin^(-1)(4/5) = sin^(-1)((12+5sqrt(7))/25)
Inverse Trigonometric Functions - Product of inverses of sine
-
Now, let’s consider an equation: sin(A)sin(B) where A and B are angles.
-
Example: Find the value of sin^(-1)(3/5)sin^(-1)(4/5).
- Using the formula, we have:
- sin^(-1)(3/5)sin^(-1)(4/5) = sin^(-1)([3/5][4/5] - sqrt(1-(3/5)^2)sqrt(1-(4/5)^2))
- sin^(-1)(3/5)sin^(-1)(4/5) = sin^(-1)(12/25 - sqrt(1-(9/25))(sqrt(1-(16/25))))
- sin^(-1)(3/5)sin^(-1)(4/5) = sin^(-1)(12/25 - sqrt(16/25-9/25))
- sin^(-1)(3/5)sin^(-1)(4/5) = sin^(-1)(12/25 - sqrt(7/25))
- sin^(-1)(3/5)sin^(-1)(4/5) = sin^(-1)(12/25 - sqrt(7)/5)
- sin^(-1)(3/5)sin^(-1)(4/5) = sin^(-1)((12-5sqrt(7))/25)
Inverse Trigonometric Functions - Difference of inverses of sine
-
Lastly, let’s consider an equation: sin(A) - sin(B) where A and B are angles.
-
Example: Find the value of sin^(-1)(3/5) - sin^(-1)(4/5).
- Using the formula, we have:
- sin^(-1)(3/5) - sin^(-1)(4/5) = sin^(-1)([3/5][4/5] - sqrt(1-(3/5)^2)sqrt(1-(4/5)^2))
- sin^(-1)(3/5) - sin^(-1)(4/5) = sin^(-1)(12/25 - sqrt(1-(9/25))(sqrt(1-(16/25))))
- sin^(-1)(3/5) - sin^(-1)(4/5) = sin^(-1)(12/25 - sqrt(16/25-9/25))
- sin^(-1)(3/5) - sin^(-1)(4/5) = sin^(-1)(12/25 - sqrt(7/25))
- sin^(-1)(3/5) - sin^(-1)(4/5) = sin^(-1)(12/25 - sqrt(7)/5)
- sin^(-1)(3/5) - sin^(-1)(4/5) = sin^(-1)((12-5sqrt(7))/25)
In summary, we have formulas for finding the sum, product, and difference of the inverses of sine. These formulas are useful when working with angles and trigonometric functions involving sine.