Let’s solve each equation one by one over the interval [0, 2π). We’ll isolate the trig function first, then find all angles in that range that satisfy it.
---
1. 2 cos x + 4 = 5
Subtract 4 from both sides:
2 cos x = 1
Divide by 2:
cos x = 1/2
Where is cosine equal to 1/2 between 0 and 2π?
→ In Quadrant I: x = π/3
→ In Quadrant IV: x = 5π/3
✔ Solutions:
x = π/3, 5π/3
---
2. 2 sin x - 1 = 0
Add 1 to both sides:
2 sin x = 1
Divide by 2:
sin x = 1/2
Where is sine equal to 1/2 between 0 and 2π?
→ In Quadrant I: x = π/6
→ In Quadrant II: x = 5π/6
✔ Solutions:
x = π/6, 5π/6
---
3. tan² x - 3 = 0
Add 3 to both sides:
tan² x = 3
Take square root of both sides:
tan x = √3 or tan x = -√3
Now find where tangent equals √3 or -√3 in [0, 2π):
- tan x = √3 → Quadrants I and III → x = π/3, 4π/3
- tan x = -√3 → Quadrants II and IV → x = 2π/3, 5π/3
✔ Solutions:
x = π/3, 2π/3, 4π/3, 5π/3
---
4. 5 cos x - √3 = 3 cos x
Subtract 3 cos x from both sides:
2 cos x - √3 = 0
Add √3 to both sides:
2 cos x = √3
Divide by 2:
cos x = √3 / 2
Where is cosine equal to √3/2 in [0, 2π)?
→ Quadrant I: x = π/6
→ Quadrant IV: x = 11π/6
✔ Solutions:
x = π/6, 11π/6
---
5. 4 csc² x - 2 = 0
Add 2 to both sides:
4 csc² x = 2
Divide by 4:
csc² x = 1/2
But wait — csc x = 1/sin x, so csc² x = 1/sin² x.
So:
1/sin² x = 1/2 → sin² x = 2
But sin² x cannot be greater than 1! The maximum value of sin² x is 1.
✘ No solution exists.
✔ Solution:
No solution
---
6. sin² x - 5 cos x = 5
We know sin² x = 1 - cos² x (Pythagorean identity)
Replace sin² x:
(1 - cos² x) - 5 cos x = 5
Simplify:
1 - cos² x - 5 cos x = 5
Move everything to one side:
- cos² x - 5 cos x + 1 - 5 = 0
→ -cos² x - 5 cos x - 4 = 0
Multiply both sides by -1 to make it easier:
cos² x + 5 cos x + 4 = 0
Factor:
(cos x + 1)(cos x + 4) = 0
Set each factor to zero:
- cos x + 1 = 0 → cos x = -1 → x = π
- cos x + 4 = 0 → cos x = -4 → Not possible! Cosine can’t be less than -1.
✔ Only solution:
x = π
---
Final Answer:
1. x = π/3, 5π/3
2. x = π/6, 5π/6
3. x = π/3, 2π/3, 4π/3, 5π/3
4. x = π/6, 11π/6
5. No solution
6. x = π
Parent Tip: Review the logic above to help your child master the concept of trigonometric equations worksheet.
