Can someone explain to me this :
-sinx=cosx
X=3pi/4 and 7pi/4
- sin x = cos x
Divide both sides by - cos x
- sin x / - cos x = cos x / - cos x
tan x = - 1
Solution:
x = pi * n - pi / 4
If n = 1 then:
x = pi * n - pi / 4 = pi * 1 - pi / 4 = pi - pi / 4 = 4 pi / 4 - pi / 4 = 3 pi / 4
If n = 2 then:
x = pi * n - pi / 4 = pi * 2 - pi / 4 = 2 pi - pi / 4 = 8 pi / 4 - pi / 4 = 7 pi / 4
To solve the equation -sin(x) = cos(x), we need to find values of x that satisfy the equation. Let's go step by step to understand how to solve it:
1. Recall the relationship between sine and cosine: sin(x) is the y-coordinate and cos(x) is the x-coordinate of a point on the unit circle at angle x.
2. Since we are solving for -sin(x) = cos(x), this means the y-coordinate of the point should be equal to the x-coordinate. This implies that the point lies on the line y = x.
3. To find the values of x that satisfy this condition, we can consider the coordinates of the points where the line y = x intersects the unit circle.
4. The unit circle is centered at the origin and has a radius of 1. The points of intersection must have coordinates (x, x) that lie on the circle.
5. From trigonometry, we know that the angle in standard position (the angle formed between the positive x-axis and the terminal side) can be found as arccos(x) or arcsin(x) depending on the y or x value, respectively.
6. Since we want to find the angles where -sin(x) = cos(x), we can set sin(x) = cos(x). Using the identity sin(x) = cos(90° - x), we have sin(x) = sin(90° - x).
7. By comparing the angles, we get x = 90° - x or x = x + 90° (since sine function repeats every 360°).
8. Solving the equation x = 90° - x gives us x = 45°, which corresponds to π/4 in radians.
9. Substituting x = π/4 into the equation sin(x) = cos(x), we find that sin(π/4) = cos(π/4), which is correct.
10. Hence, one solution is x = π/4 or 45°.
11. Now, let's solve the equation x = x + 90°. Subtracting x from both sides gives us 0 = 90°.
12. However, 0 = 90° has no valid solution because these angles do not intersect on the unit circle.
13. Therefore, the only solution for -sin(x) = cos(x) is x = π/4 or 45°.
In conclusion, the values of x that satisfy the equation -sin(x) = cos(x) are x = 3π/4 and x = 7π/4.