Question

Solve the equation

Cos 40° Cos x +sin 40° sin x for 0° less than or equal to x less than or equal to 360

Answer 1

[AI-generated answer — Bot GPT 3.5]

The given equation is:

cos(40°)cos(x) + sin(40°)sin(x)

Using the identity cos(A-B) = cosAcosB + sinAsinB, we can rewrite the equation as:

cos(40°-x)

Now, we set this equation equal to zero:

cos(40°-x) = 0

To find x in the interval 0° ≤ x ≤ 360°, we need to find the values of x that make cos(40°-x) equal to zero.

From the unit circle, we know that cos(40°-x) = 0 when the angle (40°-x) is 90° or 270°.

Set 40° - x = 90°:

40° - x = 90°
-x = 90° - 40°
-x = 50°
x = -50°

This solution is not in the given interval, so we discard it.

Set 40° - x = 270°:

40° - x = 270°
-x = 270° - 40°
-x = 230°
x = -230°

Similarly, this solution is not in the given interval, so we discard it.

Therefore, the given equation has no solution in the interval 0° ≤ x ≤ 360°.

Answer 2

[AI-generated answer — Step-by-Step Bot]

To solve the equation, we will use the trigonometric identity:

cos(a - b) = cos a cos b + sin a sin b

In this case, a = 40° and b = x. So, we have:

Cos 40° Cos x + sin 40° sin x = cos(40° - x)

Now, we need to find the value of x for which cos(40° - x) is equal to the given expression.

Since the range of x is from 0° to 360°, we need to find the values of x that make cos(40° - x) equal to cos(40°).

Using the property of cosine function that cos(a) = cos(360° - a), we have two possible solutions:

1. 40° - x = 40°
-x = 0°
x = 0°

2. 40° - x = 360° - 40°
-x = 320°
x = -320°

Therefore, the solutions to the equation Cos 40° Cos x + sin 40° sin x for 0° ≤ x ≤ 360° are x = 0° and x = -320°.

Answer 3

[AI-generated answer — Explain Bot]

To solve the equation cos 40° cos x + sin 40° sin x = 0, we can use the trigonometric identity:

cos(a - b) = cos a cos b + sin a sin b

In this equation, let a = 40° and b = x. Rearranging the identity, we get:

cos a cos b + sin a sin b = 0

cos (a - b) = 0

Now, we can solve for x by finding the values of (a - b) that yield a cosine of 0.

cos (a - b) = 0

cos (40° - x) = 0

To find the values of x for which the equation is satisfied, we want to find the solutions for which the cosine function equals 0.

cos (40° - x) = 0

To find the solutions, we need to solve for x:

40° - x = 90° + k * 180° (where k is any integer)

Solving for x:

x = 40° - 90° - k * 180°
x = - 50° - k * 180°

Therefore, the solutions for 0° ≤ x ≤ 360° are:

x = -50°, 130°, 310°, ... (k = 0, 1, 2, ...)