Evaluate each expression without using a calculator.
sin(sin^-1 1/2 + tan^-1 (-3))
To evaluate the expression sin(sin^-1(1/2) + tan^-1(-3)), we can use the trigonometric identities.
1. Let's start with sin^-1(1/2). This means, "What angle has a sine of 1/2?" The answer is π/6 or 30 degrees.
2. Next, let's consider tan^-1(-3). This means, "What angle has a tangent of -3?" The answer is approximately -1.249 radians or -71.57 degrees.
3. Now, we can add π/6 and -1.249 radians together: π/6 + (-1.249) = approximately -1.040 radians.
4. Finally, we can find the sine of -1.040 radians: sin(-1.040) ≈ -0.868.
Therefore, sin(sin^-1(1/2) + tan^-1(-3)) ≈ -0.868.