Find the value of cos 20° + cos 100° + cos 140°.

Sure, let's solve this step by step:

1. We know that cos(180° - θ) = -cos(θ). So, we can write cos 140° as -cos(180° - 140°) = -cos 40°.

2. Similarly, cos(180° - θ) = -cos(θ). So, we can write cos 100° as -cos(180° - 100°) = -cos 80°.

3. Now, the equation becomes: cos 20° - cos 80° - cos 40°.

4. We know that cos(a) - cos(b) = -2sin[(a+b)/2]sin[(a-b)/2]. So, we can write cos 20° - cos 80° as -2sin[(20°+80°)/2]sin[(80°-20°)/2] = -2sin 50° sin 30° = -sin 50°.

5. Similarly, we can write - cos 40° as -2sin[(20°+40°)/2]sin[(40°-20°)/2] = -2sin 30° sin 10° = -sin 10°.

6. So, the equation becomes: -sin 50° - sin 10°.

7. We know that sin(a) + sin(b) = 2sin[(a+b)/2]cos[(a-b)/2]. So, we can write -sin 50° - sin 10° as -2sin[(50°+10°)/2]cos[(50°-10°)/2] = -2sin 30° cos 20° = -sin 20°.

8. Therefore, cos 20° + cos 100° + cos 140° = -sin 20°.

Sure, let's solve this step by step:

1. We know that cos(180° - θ) = -cos θ. So, we can write cos 140° as -cos(180° - 140°) = -cos 40°.

2. Similarly, cos(180° + θ) = -cos θ. So, we can write cos 100° as -cos(180° - 100°) = -cos 80°.

3. Now, the equation becomes: cos 20° - cos 80° - cos 40°.

4. We know that cos(a) - cos(b) = -2sin[(a+b)/2]sin[(b-a)/2]. So, we can write cos 20° - cos 80° as -2sin[(20°+80°)/2]sin[(80°-20°)/2] = -2sin 50° sin 30° = -sin 50°.

5. Similarly, we can write cos 20° - cos 40° as -2sin[(20°+40°)/2]sin[(40°-20°)/2] = -2sin 30° sin 10° = -sin 10°.

6. So, the equation becomes: -sin 50° - sin 10°.

7. We know that sin(60° - θ) = sin 60°cos θ - cos 60°sin θ. So, we can write -sin 50° as -[sin 60°cos 10° - cos 60°sin 10°] = -√3/2 cos 10° + 1/2 sin 10°.

8. Similarly, we can write -sin 10° as -[sin 60°cos 50° - cos 60°sin 50°] = -√3/2 cos 50° + 1/2 sin 50°.

9. So, the equation becomes: -√3/2 cos 10° + 1/2 sin 10° - √3/2 cos 50° + 1/2 sin 50°.

10. We can simplify this to: -√3/2 (cos 10° + cos 50°) + 1/2 (sin 10° + sin 50°).

11. We know that sin(θ + 45°) = √2/2(sin θ + cos θ). So, we can write sin 10° + cos 10° as √2 sin(10° + 45°) = √2 sin 55°.

12. Similarly, we can write sin 50° + cos 50° as √2 sin(50° + 45°) = √2 sin 95°.

13. So, the equation becomes: -√3/2 √2 sin 55° + 1/2 √2 sin 95°.

14. We can simplify this to: -√6/4 sin 55° + √2/4 sin 95°.

15. So, the value of cos 20° + cos 100° + cos 140° is -√6/4 sin 55° + √2/4 sin 95°.