Solve 3cos2x - 7cosx = 0, when 0<=x<=360

And also, find the exact values for x when 0<=x<=360 if 3tan^2x=1

Thankyou!

have you come across

cos 2x = 2cos^2 x - 1 ?
let's sub it.
3cos 2x - 7cosx = 0
3(2cos^2 x - 1) - 7cosx = 0
6cos^2x - 7cosx - 3 = 0
(2cosx-3)(3cosx+1) = 0

cosx = 3/2 , which is not possible,
or
cosx = -1/3
so x must be in quadrants II or III
angle in standard position is 7.53 , then
x = 180-70.53 = 109.47°
or
x = 180+70.53 = 250.53

3tan^2 = 1

tan^2x = 1/3
tanx = ± 1/√3

x = 30°, 150°, 210° or 330°

3cos2x+7cosx+3=0

To solve the equation 3cos2x - 7cosx = 0, we can use trigonometric identities and algebraic techniques. Here's how:

Step 1: Express cos2x in terms of cosx. We know that cos2x = cos^2x - sin^2x. Since sin^2x = 1 - cos^2x (from the Pythagorean identity), we can substitute this in:
cos2x = cos^2x - (1 - cos^2x)
cos2x = 2cos^2x - 1

Step 2: Substitute this expression into the equation:
3(2cos^2x - 1) - 7cosx = 0

Step 3: Simplify the equation:
6cos^2x - 3 - 7cosx = 0
6cos^2x - 7cosx - 3 = 0

Step 4: Solve the quadratic equation. This can be factored or solved using the quadratic formula. In this case, factoring is not straightforward, so we will use the quadratic formula:
cosx = (-(-7) ± √((-7)^2 - 4(6)(-3)))/(2(6))
cosx = (7 ± √(49 + 72))/12
cosx = (7 ± √121)/12
cosx = (7 ± 11)/12

Step 5: Simplify and find the values of cosx. We have two possibilities:
(i) cosx = (7 + 11)/12 = 18/12 = 3/2 > 1 (not within the range of possible values)
(ii) cosx = (7 - 11)/12 = -4/12 = -1/3

Step 6: Find the values of x within the given range. Recall that cosx = -1/3. We can use the inverse cosine function to find the angles in degrees:
x = arccos(-1/3) ≈ 109.47 degrees (approximately)

Please note that the equation has only one valid solution based on the given range (0 <= x <= 360), which is approximately x = 109.47 degrees.

Now let's move on to the second question.

To find the exact values for x when 0 <= x <= 360, given that 3tan^2x = 1, we can follow these steps:

Step 1: Solve the equation for tan(x):
tan^2x = 1/3

Step 2: Take the square root of both sides:
tanx = ± √(1/3)

Step 3: Use the inverse tangent function to find the angles in degrees:
x = arctan(± √(1/3))

Step 4: Calculate the value:
x ≈ ± 30 degrees, ± 150 degrees

Therefore, the exact values for x when 0 <= x <= 360 when 3tan^2x = 1 are approximately x = 30 degrees, 150 degrees, 210 degrees, and 330 degrees.