 #1
AN630078
 242
 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
 Relevant Equations:
 π
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.
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.