sin(((cos^-1)(3/5))-((sin^-1)(5/13)))
help and explain please
let A = cos^-1 (3/5)
then cos A = 3/5, and sinA = 4/5 (using the 3-4-5 triangle)
let B = sin^-1 (5/13)
sin B = 5/13 and cosB = 12/13
so sin(((cos^-1)(3/5))-((sin^-1)(5/13)))
= sin(A - B)
= sinAcosB - cosAsinB
= (4/5)(12/13) - (3/5)(5/13)
= 48/64 - 15/64
= 33/65
thanks a lot
it helped
(:
To simplify the given expression, let's break it down step by step.
Step 1: Evaluate the innermost function, (cos^-1)(3/5).
The notation (cos^-1)(x) represents the inverse cosine function, also known as arccosine, which returns the angle whose cosine is x.
So, (cos^-1)(3/5) means finding the angle whose cosine is 3/5.
Step 2: Evaluate the next innermost function, (sin^-1)(5/13).
The notation (sin^-1)(x) represents the inverse sine function, also known as arcsine, which returns the angle whose sine is x.
So, (sin^-1)(5/13) means finding the angle whose sine is 5/13.
Step 3: Substitute the evaluated values into the expression.
sin(((cos^-1)(3/5))-((sin^-1)(5/13))) becomes sin(A - B), where A is the angle whose cosine is 3/5 and B is the angle whose sine is 5/13.
Step 4: Use the trigonometric identity sin(A - B) = sin(A)cos(B) - cos(A)sin(B).
sin(((cos^-1)(3/5))-((sin^-1)(5/13))) becomes sin(A)cos(B) - cos(A)sin(B), where A is the angle whose cosine is 3/5 and B is the angle whose sine is 5/13.
Step 5: Evaluate sin(A), cos(A), sin(B), and cos(B).
To find the values of sin(A), cos(A), sin(B), and cos(B), you will need a calculator or a trigonometric table. Using the known values of A and B, you can calculate these trigonometric functions.
For example, if A = cos^-1(3/5) = 53.13° and B = sin^-1(5/13) = 22.62°, you can approximate the values of sin(A), cos(A), sin(B), and cos(B) using a calculator.
Step 6: Plug in the values and calculate the final answer.
Once you have calculated the values of sin(A), cos(A), sin(B), and cos(B), substitute them into the expression sin(A)cos(B) - cos(A)sin(B) and calculate the final answer.
Keep in mind that without the exact values of A and B, the final answer will be an approximation.