Question

√3 sin x — cos x = 0

Answer 1

Ответ:

π/6

Объяснение:

$\sqrt{3} \sin(x) - \cos(x) = 0 \\ \sqrt{3} \sin(x ) = \cos(x) \: \: | \div \cos(x) \\ \sqrt{3} \tan(x) = 1 \: \: | \div \sqrt{3} \\ \tan(x) = \frac{1}{ \sqrt{3} } \\ \tan(x) = \frac{ \sqrt{3} }{3} \\ x = arctan( \frac{ \sqrt{3} }{3} ) \\ x = \frac{\pi}{6}$

Answer 2

To solve the equation √3 sin(x) - cos(x) = 0, you can use trigonometric identities and algebraic manipulations. Here's how you can do it:

1. Start by dividing both sides of the equation by √3 to simplify it:

   sin(x) - (1/√3) cos(x) = 0

2. Now, use the identity sin(π/6) = 1/2 and cos(π/6) = √3/2, which are values of the sine and cosine functions at 30 degrees:

   sin(π/6) = 1/2 cos(π/6) = √3/2

3. Multiply both sides of the equation by 2 to eliminate the fractions:

   2sin(x) - 2(1/√3)cos(x) = 0

4. Now, rewrite the equation using these values:

   2sin(x) - (2/√3)cos(x) = 0

5. Use the sum-to-product trigonometric identity for sin(a)cos(b):

   2sin(x) - (2/√3)cos(x) = 2sin(x) - 2sin(π/6)cos(x) - 2cos(π/6)sin(x) = 0

6. Combine like terms:

   2sin(x) - (2/√3)cos(x) - (2/√3)cos(x) = 0

7. Factor out sin(x) from the first two terms:

   2sin(x)(1 - 1/√3) - (2/√3)cos(x) = 0

8. Now, solve for sin(x):

   2sin(x)(1 - 1/√3) = (2/√3)cos(x)

9. Divide both sides by 2(1 - 1/√3):

   sin(x) = (2/√3)cos(x) / [2(1 - 1/√3)]

10. Now, use the fact that sin(x) = cos(π/2 - x) and cos(x) = sin(π/2 - x):

sin(x) = cos(π/2 - x) sin(x) = sin(π/2 - x)

11. Set the angles equal to each other:

x = π/2 - x

12. Solve for x:

2x = π/2

x = π/4

So, the solution to the equation √3 sin(x) - cos(x) = 0 is x = π/4.