How do you simplify
tan(sin^-1 (x)) ?
draw a triangle, hypotenuse 1, opposite side x, adjacent side sqrt(1-x^2)
tan of that angle is x/sqrt(1-x^2)
your instructor is just to easy.
How did you come to those triangle dimensions?
sin = x/1, x being opposite, 1 being hypotenuse......................................................
To simplify the expression tan(sin⁻¹(x)), we can use trigonometric identities.
Step 1: Recall the definition of inverse trigonometric functions. The expression sin⁻¹(x) represents the angle whose sine is x.
Step 2: Let's assume the angle whose sine is x to be θ. Therefore, sin(θ) = x. Now, we need to find the tangent of this angle.
Step 3: To find the tangent of an angle, we can use the identity tan(θ) = sin(θ) / cos(θ).
Step 4: We already know that sin(θ) = x. Now, we need to find cos(θ) to complete our simplification.
Step 5: Using the Pythagorean identity sin²(θ) + cos²(θ) = 1, we can solve for cos(θ). Since sin(θ) = x, we have x² + cos²(θ) = 1.
Step 6: Rearranging the equation, we have cos²(θ) = 1 - x². Taking the square root of both sides, we get cos(θ) = ±√(1 - x²).
Step 7: Now, we substitute the values of sin(θ) = x and cos(θ) = ±√(1 - x²) into the formula for tangent: tan(θ) = x / ±√(1 - x²).
Step 8: Simplifying further, we can rewrite the expression as tan(sin⁻¹(x)) = x / ±√(1 - x²).
So, the simplified expression for tan(sin⁻¹(x)) is x / ±√(1 - x²), where ± indicates that the positive as well as the negative values should be considered.