Find all solutions for arccos(x) – arcsin(x) = pi/6.
The Correct Answer and Explanation is:
We are tasked with finding all solutions for the equation:arccos(x)−arcsin(x)=π6\arccos(x) – \arcsin(x) = \frac{\pi}{6}arccos(x)−arcsin(x)=6π
Step 1: Use a trigonometric identity
We start by using the identity for inverse trigonometric functions. Specifically, we know that:arccos(x)+arcsin(x)=π2\arccos(x) + \arcsin(x) = \frac{\pi}{2}arccos(x)+arcsin(x)=2π
This identity holds for any value of xxx within the domain [−1,1][-1, 1][−1,1], since the arccosine and arcsine functions are related in this way.
Step 2: Express arccos(x)\arccos(x)arccos(x) in terms of arcsin(x)\arcsin(x)arcsin(x)
From the identity above, we can solve for arccos(x)\arccos(x)arccos(x):arccos(x)=π2−arcsin(x)\arccos(x) = \frac{\pi}{2} – \arcsin(x)arccos(x)=2π−arcsin(x)
Step 3: Substitute into the given equation
Now, substitute arccos(x)=π2−arcsin(x)\arccos(x) = \frac{\pi}{2} – \arcsin(x)arccos(x)=2π−arcsin(x) into the original equation:(π2−arcsin(x))−arcsin(x)=π6\left( \frac{\pi}{2} – \arcsin(x) \right) – \arcsin(x) = \frac{\pi}{6}(2π−arcsin(x))−arcsin(x)=6π
Simplify this equation:π2−2arcsin(x)=π6\frac{\pi}{2} – 2\arcsin(x) = \frac{\pi}{6}2π−2arcsin(x)=6π
Step 4: Solve for arcsin(x)\arcsin(x)arcsin(x)
Now, isolate the term involving arcsin(x)\arcsin(x)arcsin(x):−2arcsin(x)=π6−π2-2\arcsin(x) = \frac{\pi}{6} – \frac{\pi}{2}−2arcsin(x)=6π−2π
First, simplify the right side:π6−π2=π6−3π6=−2π6=−π3\frac{\pi}{6} – \frac{\pi}{2} = \frac{\pi}{6} – \frac{3\pi}{6} = -\frac{2\pi}{6} = -\frac{\pi}{3}6π−2π=6π−63π=−62π=−3π
Now the equation becomes:−2arcsin(x)=−π3-2\arcsin(x) = -\frac{\pi}{3}−2arcsin(x)=−3π
Dividing both sides by -2:arcsin(x)=π6\arcsin(x) = \frac{\pi}{6}arcsin(x)=6π
Step 5: Solve for xxx
Finally, solve for xxx by taking the sine of both sides:x=sin(π6)x = \sin\left(\frac{\pi}{6}\right)x=sin(6π)
Since sin(π6)=12\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}sin(6π)=21, we have:x=12x = \frac{1}{2}x=21
Step 6: Verify the solution
Let’s verify the solution by checking that x=12x = \frac{1}{2}x=21 satisfies the original equation.
- arccos(12)=π3\arccos\left(\frac{1}{2}\right) = \frac{\pi}{3}arccos(21)=3π
- arcsin(12)=π6\arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6}arcsin(21)=6π
Substituting these into the original equation:arccos(12)−arcsin(12)=π3−π6=π6\arccos\left(\frac{1}{2}\right) – \arcsin\left(\frac{1}{2}\right) = \frac{\pi}{3} – \frac{\pi}{6} = \frac{\pi}{6}arccos(21)−arcsin(21)=3π−6π=6π
This is true, so the solution is correct.
Final Answer:
The solution to the equation arccos(x)−arcsin(x)=π6\arccos(x) – \arcsin(x) = \frac{\pi}{6}arccos(x)−arcsin(x)=6π is:x=12x = \frac{1}{2}x=21
