Find all the angles between 0 degrees and 360 degrees which satisfy the equations
a. 8 cos x sin x = sin x
b. 5 tan^2 y + 5 tan y = 2 sec^2 y.
b)
5 tan^2 y + 5 tan y = 2 sec^2 y
playing around with some right-angled triangles, using the numbers 2 and 5
I drew a triangle with base 1, height 2 and hypotenuse 5
let the base angle be y
then tany = 2/1 , y = 63.4349...° ( stored in memory)
checking:
LS = 5(2/1)^2 + 5(2/1) = 30
RS = 2/(1/√5)^2 = 10
ahh, but if tany = -2/1
LS = 5(-2/1)^2 + 5(-2/1) = 10
RS = 1/(1/√5)^2 = 10
so we need y = 180-63.4349 or y = 360-63.4349
y = 116.565° or y = 296.565°
a. To solve the equation 8 cos x sin x = sin x, we can rearrange it as follows:
8 cos x sin x - sin x = 0
We can factor out sin x from the left side of the equation:
sin x(8 cos x - 1) = 0
Now we have two possibilities for sin x to be equal to zero:
1. sin x = 0, which means x = 0 degrees or x = 180 degrees.
Now let's solve for the second factor:
8 cos x - 1 = 0
Add 1 to both sides:
8 cos x = 1
Divide both sides by 8:
cos x = 1/8
Now, we can find the angles between 0 degrees and 360 degrees that satisfy this equation by taking the inverse cosine or arccosine of 1/8:
x = arccos(1/8) ≈ 84.6 degrees or x = -arccos(1/8) ≈ 275.4 degrees
Therefore, the angles between 0 degrees and 360 degrees that satisfy the equation 8 cos x sin x = sin x are:
x = 0 degrees, 180 degrees, 84.6 degrees, and 275.4 degrees.
b. To solve the equation 5 tan^2 y + 5 tan y = 2 sec^2 y, we can begin by expressing sec^2 y in terms of tan y:
sec^2 y = 1 + tan^2 y
Substituting this into the original equation:
5 tan^2 y + 5 tan y = 2(1 + tan^2 y)
Distribute the 2:
5 tan^2 y + 5 tan y = 2 + 2 tan^2 y
Rearrange the equation by moving all the terms to one side:
3 tan^2 y + 5 tan y - 2 = 0
This is a quadratic equation in terms of tan y. We can solve it by factoring or using the quadratic formula. In this case, let's use factoring.
The equation can be factored as follows:
(3 tan y - 1)(tan y + 2) = 0
Setting each factor equal to zero, we have:
3 tan y - 1 = 0 or tan y + 2 = 0
For 3 tan y - 1 = 0, solve for tan y:
3 tan y = 1
tan y = 1/3
Taking the inverse tangent or arctangent of 1/3:
y = arctan(1/3) ≈ 18.4 degrees
For tan y + 2 = 0, solve for tan y:
tan y = -2
Taking the inverse tangent or arctangent of -2:
y = arctan(-2) ≈ -63.4 degrees
Therefore, the angles between 0 degrees and 360 degrees that satisfy the equation 5 tan^2 y + 5 tan y = 2 sec^2 y are:
y = 18.4 degrees and -63.4 degrees.
To find the angles that satisfy the given equations, we need to solve the equations and find the values of x and y that make the equations true. Let's solve the equations one by one:
a. 8 cos x sin x = sin x
Step 1: Divide both sides by sin x:
8 cos x = 1
Step 2: Divide both sides by 8:
cos x = 1/8
Step 3: Take the inverse cosine (cos^(-1)) of both sides to find the angle x:
x = cos^(-1)(1/8)
Using a calculator, evaluate the inverse cosine of 1/8 to find the angle x. This will give you a value between 0 and π radians or 0 and 180 degrees.
b. 5 tan^2 y + 5 tan y = 2 sec^2 y
Step 1: Rewrite sec^2 y in terms of tan y using the identity sec^2 y = 1 + tan^2 y:
5 tan^2 y + 5 tan y = 2 (1 + tan^2 y)
Step 2: Distribute 2 to both terms on the right side:
5 tan^2 y + 5 tan y = 2 + 2 tan^2 y
Step 3: Subtract 2 tan^2 y from both sides to isolate the terms with tan y:
3 tan^2 y + 5 tan y - 2 = 0
Step 4: Factorize the quadratic equation:
(3 tan y - 1)(tan y + 2) = 0
Step 5: Set each factor equal to zero and solve for y:
3 tan y - 1 = 0 or tan y + 2 = 0
For the first equation, add 1 to both sides and divide by 3 to solve for tan y:
tan y = 1/3
For the second equation, subtract 2 from both sides and solve for tan y:
tan y = -2
Using a calculator, take the inverse tangent (tan^(-1)) of 1/3 and -2 to find the angles y. This will give you values between -π/2 and π/2 radians or -90 and 90 degrees.
So, by following these steps, you can find the angles between 0 degrees and 360 degrees that satisfy the given equations.
a)
8cosxsinx - sinx = 0
sinx(8cosx - 1) = 0
sinx = 0 or cosx = 1/8
if sinx = 0 , x = 0, 180, 360
if cosx = 1/8, x = 82.8° or 277.18°
working on the 2nd....