Problem 68 - Precalculus: A Concise Course

Question

Verify that \(\sin \left(t_{1}+t_{2}\right) \neq \sin t_{1}+\sin t_{2}\) by approximating \(\sin 0.25, \sin 0.75,\) and \(\sin 1\).

Step-by-Step Solution

Answer

Upon calculation, it's seen that sin(t1+t2) is not equal to sin(t1) + sin(t2), even using approximations. So, the statement was verified to be true.

Step 1: Calculate Individual Sine Values

First, we need to compute the sine values for 0.25, 0.75 , and 1. You can do this using a calculator, in radians.

Step 2: Apply Addition Formula on Left Hand Side

Now, plug in the values of t1 and t2 into the left side of the equation, which is sin(t1+t2). Here, we assumed t1 is 0.25 and t2 is 0.75, hence calculate sin(0.25+0.75).

Step 3: Sum up Sine Values on Right Hand Side

Then we add the sine values of t1 and t2 for the right side of the equation, which is sin(t1)+sin(t2). That will be sin(0.25) + sin(0.75).

Step 4: Compare Both Sides

Now we compare the value we got in step 2: sin(t1+t2) with the value we got in step 3: sin(t1) + sin(t2).