- #1

AN630078

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- 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 (2θ-π/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(π-θ)

π-π/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(2π-θ)

2π-π/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 (2θ-π/4)=1

2θ-π/4=arctan 1

2θ-π/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(π-θ)

π-π/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(2π-θ)

2π-π/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 (2θ-π/4)=1

2θ-π/4=arctan 1

2θ-π/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.