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 25
 Homework Statement:

Trigonometric equations are by far one of my weakest areas. I have been practising to improve and refine my understanding but I am still a little uncertain in areas. I have attempted some questions below but was wondering if anyone could offer me some advice on how to improve my workings or apply more suitable methods.
Question 1; Solve the following equations giving your solutions as exact fractions of π in the range 0 ≤θ ≤2 π:
a. sin θ=√3/2
b.cos2θ=0.5
c.tan (2θπ/4)=1
Question 2; Sketch the graphs y=2cos x and y=tan x.
a. How many values of x satisfy the equation in the range 0 ≤θ ≤360 degrees?
b. Show that the xvalues at the points of intersection satisfy the equation 2sin^2x+sins2=0
c. Solve the equation 2cos x=tan x between 0 and 360 degrees
 Relevant Equations:

tan x=sin x / cos x
sin^2x+cos^2x=1
Question 1;
a. sin θ=√3/2
θ=arcsin √3/2
θ=π/3 rad
sin √3/2=60 degrees
60 degrees *π/180=π/3 rad.
To find the other solutions in the range, sin θ=sin(πθ)
ππ/3=2π/3
The solutions are π/3 and 2π/3 in the range 0 ≤θ ≤2 π
b. cos2θ=0.5
2θ=arccos 0.5
2θ=π/3 rad
Divide both sides by 2;
θ=π/6 rad
To find the other solutions in the range, cos θ=cos(2πθ)
2ππ/3=5π/3
Divide by 2 =5π/6
The solutions are π/6 and 5π/6 in the range 0 ≤θ ≤2 π
c. tan (2θπ/4)=1
2θπ/4=arctan 1
2θπ/4 = π/4
Add π/4 to both sides;
2θ=π/4+π/4
2θ=π/2
Divide both sides by 2:
θ=π/4
To find other solutions in the range add π/2:
π/4+π/2=3π/4
3π/4+π/2=5π/4
5π/4+π/2=7π/4
The solutions are π/4, 3π/4, 5π/4 and 7π/4 in the range 0 ≤θ ≤2 π
I have also attached what I think the graphs of these equations would look like to find the required solutions. How can I improve or broaden my answers to more extensively exhibit my workings. I am a little confused here admittedly.
Question 2;
a. I have just plotted the graph using desmos and attached an image here. Clearly, there are two values of x that satisfy the equation in the range. Do I need to add anything to this statement, I feel the response is a little brief for the question?
b. Using the trigonometric identities;
tan x=sin x / cos x
and sin^2x+cos^2x=1
2cos x=tan x
Multiply the whole equation by cos x:
2cos x=sinx
Using the identity sin^2x+cos^2x=1, 2cos x becomes: 2(1sin^2x)
2(1sin^2x)=sinx
Expand the brackets;
22sin^2x=sinx
Subtract sin x from both sides;
22sin^2xsinx=0
Divide by 1;
2sin^2x+sinx2=0
c. To solve the equation 2cos x=tan x
This is shown to be equal to the quadratic;
2sin^2x+sinx2=0
Let sin x = u
2u^2+u2=0
Using the quadratic formula; a=2, b=1, c=2
u=b±√b^24ac/2a
u=1±√1^24*2*(2)/2*2
u=1±√17/4
u=0.780 and x=1.28 to 3.s.f
sin x = 1±√17/4
x=arcsin 0.78 =51.3 degrees to 3.s.f
x= arcsin 1.28 = no real solutions as x cannot be smaller than 1 for real solutions
To find other solutions in the range 0 ≤θ ≤360 degrees use sin θ=sin(180θ):
18051.3=128.7 degrees
So the solutions are 51.3 and 129 degrees to 3.s.f?
Would this be correct. I am very uncertain of trigonometric equation problems and I have been teaching myself which is perhaps why my knowledge is a little unstable and I am am sure in places erroneous. I would be very grateful of any help
a. sin θ=√3/2
θ=arcsin √3/2
θ=π/3 rad
sin √3/2=60 degrees
60 degrees *π/180=π/3 rad.
To find the other solutions in the range, sin θ=sin(πθ)
ππ/3=2π/3
The solutions are π/3 and 2π/3 in the range 0 ≤θ ≤2 π
b. cos2θ=0.5
2θ=arccos 0.5
2θ=π/3 rad
Divide both sides by 2;
θ=π/6 rad
To find the other solutions in the range, cos θ=cos(2πθ)
2ππ/3=5π/3
Divide by 2 =5π/6
The solutions are π/6 and 5π/6 in the range 0 ≤θ ≤2 π
c. tan (2θπ/4)=1
2θπ/4=arctan 1
2θπ/4 = π/4
Add π/4 to both sides;
2θ=π/4+π/4
2θ=π/2
Divide both sides by 2:
θ=π/4
To find other solutions in the range add π/2:
π/4+π/2=3π/4
3π/4+π/2=5π/4
5π/4+π/2=7π/4
The solutions are π/4, 3π/4, 5π/4 and 7π/4 in the range 0 ≤θ ≤2 π
I have also attached what I think the graphs of these equations would look like to find the required solutions. How can I improve or broaden my answers to more extensively exhibit my workings. I am a little confused here admittedly.
Question 2;
a. I have just plotted the graph using desmos and attached an image here. Clearly, there are two values of x that satisfy the equation in the range. Do I need to add anything to this statement, I feel the response is a little brief for the question?
b. Using the trigonometric identities;
tan x=sin x / cos x
and sin^2x+cos^2x=1
2cos x=tan x
Multiply the whole equation by cos x:
2cos x=sinx
Using the identity sin^2x+cos^2x=1, 2cos x becomes: 2(1sin^2x)
2(1sin^2x)=sinx
Expand the brackets;
22sin^2x=sinx
Subtract sin x from both sides;
22sin^2xsinx=0
Divide by 1;
2sin^2x+sinx2=0
c. To solve the equation 2cos x=tan x
This is shown to be equal to the quadratic;
2sin^2x+sinx2=0
Let sin x = u
2u^2+u2=0
Using the quadratic formula; a=2, b=1, c=2
u=b±√b^24ac/2a
u=1±√1^24*2*(2)/2*2
u=1±√17/4
u=0.780 and x=1.28 to 3.s.f
sin x = 1±√17/4
x=arcsin 0.78 =51.3 degrees to 3.s.f
x= arcsin 1.28 = no real solutions as x cannot be smaller than 1 for real solutions
To find other solutions in the range 0 ≤θ ≤360 degrees use sin θ=sin(180θ):
18051.3=128.7 degrees
So the solutions are 51.3 and 129 degrees to 3.s.f?
Would this be correct. I am very uncertain of trigonometric equation problems and I have been teaching myself which is perhaps why my knowledge is a little unstable and I am am sure in places erroneous. I would be very grateful of any help