Evaluate the expression : tan(sin^-1(9/13)).
How would I do this without using a calculator to find the inverse values?
draw a right triangle with one leg=9 and hypotenuse=13
so, the other leg is √88
now, the tangent of the angle whose sine is 9/13 is 9/√88
start with sin^-1(9/13)
to me that means tanØ = 9/13, that is, I have a right-angled triangle where the hypotenuse is 13 and the opposite side is 9, with a base angle of Ø
to find the adjacent: x^2 + 9^2 = 13^2
x^2 = 169 - 81 = 88
x = √88
so now we need tanØ = y/x = 9/√88
tan(sin^-1(9/13)) = 9/√88 , simplify this radical if you have to.
To evaluate the expression tan(sin⁻¹(9/13)), we need to break it down step by step. Let's start with the innermost function, sin⁻¹(9/13).
1. sin⁻¹(9/13) represents the inverse sine function. It gives us the angle whose sine is equal to 9/13. To evaluate this, we can use a calculator or a trigonometric table.
Using a calculator, you can press the sin⁻¹ button or arcsin button, enter 9/13 (or 0.6923076923076923 approximately), and find the angle to be 43.60 degrees (approximately).
2. Now that we know that sin⁻¹(9/13) is equal to 43.60 degrees, we can substitute this value back into the original expression.
tan(43.60 degrees)
3. Finally, we evaluate the tangent function with the angle 43.60 degrees. Again, you can use a calculator or built-in functions to find the tangent of this angle.
Using a calculator, input the value 43.60, press the tan button, and you will find that tan(43.60 degrees) is approximately 0.9325.
Therefore, the value of the expression tan(sin⁻¹(9/13)) is approximately 0.9325.