Solve the equation
Cos40° Cosx + sin 40° sin x=square root 3/2
Using the identity Cos(A-B) = CosA CosB + SinA SinB, observe that your equation can be rewritten by rearranging letters in the identity to Cos(40° - x) = sqrt(3)/2.
Now, using a special angles in trigonometry, the angle whose cosine equals the square root of 3 divided by 2 is 30 degrees or π/6 (in radians).
Therefore, the solutions are given by:
40° - x = 30°, which simplifies to x = 10°
AND
40° - x = 30° + 180k for any integer k.
Therefore, x = 40° - 30° + 180k = 10° + 180k for any integer k.
To solve the equation, we will use the trigonometric identity for the cosine of the difference of two angles:
cos(A - B) = cos A * cos B + sin A * sin B
Comparing the given equation to the identity, we can see that:
A = 40°
B = x
cos A = cos 40°
cos B = cos x
sin A = sin 40°
sin B = sin x
Now, let's substitute these values into the identity:
cos(A - B) = cos A * cos B + sin A * sin B
cos(40° - x) = cos 40° * cos x + sin 40° * sin x
Substituting the given values into the equation, we have:
cos(40° - x) = cos 40° * cos x + sin 40° * sin x
cos(40° - x) = cos 40° * cos x + sin 40° * sin x = √3/2
Now, we can simplify the equation:
cos(40° - x) = √3/2
To solve for x, we need to find the angle whose cosine is √3/2. Since the cosine of 60° is √3/2, we can conclude that 40° - x = 60°.
Subtracting 40° from both sides of the equation:
-x = 60° - 40°
-x = 20°
Finally, multiplying both sides by -1 to isolate x:
x = -20°
Therefore, the solution to the equation is x = -20°.
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).
In the given equation, we have cos(40°)cos(x) + sin(40°)sin(x). We notice that this matches with the identity cos(A - B), where A = 40° and B = x.
So, we can rewrite the equation as: cos(40° - x) = √3/2.
Now, we need to find the value of (40° - x) that satisfies this equation.
To solve for x, we'll take the inverse cosine (or arc cosine) of (√3/2) on both sides of the equation:
40° - x = arccos(√3/2).
To isolate x, simply subtract 40° from both sides of the equation:
x = 40° - arccos(√3/2).
Note that arccos returns an angle within the range of 0° to 180°, so make sure to consider the quadrant of the original angle to obtain the correct value for x.
Therefore, x = 40° - arccos(√3/2).