Solve the equation
Cos40° Cosx + sin 40° sin x=square root 3/2 for 0° less than or equal to x less than or equal to 360°
The given equation, cos40° cosx + sin 40° sin x, can be simplified using the formula for the cosine of a difference of two angles: cos(A - B) = cosA cosB + sinA sinB.
This gives us that our equation is equivalent to cos(x - 40°) = √3/2.
Now we need to solve for x in this new equation.
The angles for which cosine is √3/2 are 30° and 330° (in the range from 0° to 360°). Therefore, we have:
x - 40° = 30°
=> x = 70°
and
x - 40° = 330°
=> x = 370°
However, since we want 0° ≤ x ≤ 360°, the solution x = 370° is not in the feasible range.
Therefore, the solution to the equation is x = 70°.
To solve the equation cos 40° cos x + sin 40° sin x = √3/2, we will use the trigonometric identity for the cosine of a difference of angles.
cos(A - B) = cos A cos B + sin A sin B
First, let's rewrite the equation using the identity mentioned above:
cos 40° cos x + sin 40° sin x = √3/2
cos(40° - x) = √3/2
Now, we need to find the values of x that satisfy this equation. To do that, we can take the inverse cosine of both sides:
40° - x = ±60° + 360°n [n is an integer]
-x = ±60° - 40° + 360°n
-x = ±20° + 360°n
Now, we can solve for x:
Case 1:
-x = 20° + 360°n
x = -20° - 360°n
Case 2:
-x = -20° + 360°n
x = 20° + 360°n
Thus, the solution for 0° ≤ x ≤ 360° is:
x = -20° - 360°n, 20° + 360°n [where n is an integer]
To solve the equation cos(40°)cos(x) + sin(40°)sin(x) = √3/2, we can use the trigonometric identity:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B).
Rearranging the equation, we have:
cos(x - 40°) = √3/2.
The value of √3/2 corresponds to an angle of 30° in the unit circle. Therefore, we need to solve the equation cos(x - 40°) = cos(30°).
Using the identity cos(A) = cos(-A), we can rewrite the equation as:
x - 40° = ±30° + 360°n, where n is an integer.
To solve for x, we add 40° to both sides:
x = 40° ± 30° + 360°n.
Now, we can solve for x by substituting different values of n into the equation:
For n = 0:
x = 40° ± 30° = 70° or 10°.
For n = 1:
x = 40° ± 30° + 360° = 400° or 340°.
Thus, the possible values of x, between 0° and 360°, that satisfy the equation are: 10°, 70°, 340°, and 400°.