I have some trigonometric equations to do, but I'm pretty lost, and I have to get them done in a timely fashion, so any help would be much appreciated.
"Solve the following trig equations. Give all the positive values of the angle between 0 degrees and 360 degrees that will satisfy each. Give any approximate value to the nearest minute only."
1. sin 2x = (sqrt 3)/2
2. sin 2x - cos x = 0
3. sin (2x - 10degrees) = 1/2
4. cos 2x - sin^2 x/2 + 3/4 = 0
5. sin^2 x = cos^2 x + 1/2
6. cos 4x = sin 2x
7. 3sin x - 4cos x = 2
8. tan (x + 15degrees) = 3tan x
I know it's a lot, but even if you don't do all of them I'd be thankful.
Also, please show your work; the more detail, the better. If I can see what you're doing in detail it will help me understand.
Certainly! I'll provide step-by-step explanations for each trigonometric equation. Let's solve them one by one:
1. sin 2x = (sqrt 3)/2
To solve this equation, we need to find the values of x that satisfy the equation between 0 degrees and 360 degrees. We can rewrite the equation using the inverse sine function:
2x = arcsin((sqrt 3)/2)
Let's find the reference angle first:
reference angle = arcsin(sqrt 3) = 60 degrees
Now, we need to find the angles in the first and second quadrants that have a reference angle of 60 degrees:
x = (reference angle)/2 = 30 degrees (in the first quadrant)
2x = 60 degrees (in the second quadrant)
So, the positive angles that satisfy the equation are x = 30 degrees and x = 60 degrees.
2. sin 2x - cos x = 0
Start by rewriting the equation:
2sin x cos x - cos x = 0
Factor out cos x:
cos x (2sin x - 1) = 0
Now, set each factor equal to zero and solve for x:
cos x = 0 ---> x = 90 degrees, 270 degrees
2sin x - 1 = 0 ---> sin x = 1/2 ---> x = 30 degrees, 150 degrees, 210 degrees, 330 degrees
The positive angles that satisfy the equation are x = 30 degrees, 150 degrees, 210 degrees, and 330 degrees.
3. sin (2x - 10degrees) = 1/2
Start by finding the reference angle:
reference angle = arcsin(1/2) = 30 degrees
Now, solve for 2x - 10 degrees:
2x - 10 degrees = reference angle
2x = reference angle + 10 degrees
x = (reference angle + 10 degrees)/2
Substitute the reference angle:
x = (30 degrees + 10 degrees)/2 = 20 degrees
The positive angle that satisfies the equation is x = 20 degrees.
4. cos 2x - sin^2 (x/2) + 3/4 = 0
To simplify this equation, let's take it one step at a time:
cos 2x - sin^2 (x/2) + 3/4 = 0
Using the double-angle identity for cosine:
1 - 2sin^2 x + 3/4 = 0
Multiply everything by 4 to get rid of the fraction:
4 - 8sin^2 x + 3 = 0
Rearranging the equation:
8sin^2 x = 7
sin^2 x = 7/8
Taking the square root:
sin x = sqrt(7)/2
Find the reference angle:
reference angle = arcsin(sqrt(7)/2)
Now, solve for x:
x = reference angle = 46.56 degrees (rounded to the nearest minute)
The positive angle that satisfies the equation is x = 46 degrees and 34 minutes.
I'll continue with the remaining equations in the next response to keep the answer within the character limit.