sin(x - 20)° = cos(5x - 10)°
Expand 5x - 10 in terms of x - 20:
5x - 10 = 5x - 100 + 90
… = 5 (x - 20) + 90
and recall that for all θ, cos(θ + 90)° = - sin(θ)°, so
sin(x - 20)° = cos(5 (x - 20) + 90)°
sin(x - 20)° = - sin(5 (x - 20))°
Replace (x - 20)° = y ° to make this a bit easier to read:
sin(y °) = - sin(5y )°
Expand the right side in terms of powers of sin:
sin(5θ) = 5 sin(θ) - 20 sin³(θ) + 16 sin⁵(θ)
so the equation becomes
sin(y °) = -5 sin(y °) + 20 sin³(y °) - 16 sin⁵(y °)
16 sin⁵(y °) - 20 sin³(y °) + 6 sin(y °) = 0
Replace z = sin(y °) to get a polynomial equation:
16z ⁵ - 20z ³ + 6z = 0
Factorize the left side:
2z (8z ⁴ - 10z ² + 3) = 0
2z (2z ² - 1) (4z ² - 3) = 0
Solve for z :
• 2z = 0 → z = 0
• 2z ² - 1 = 0 → z ² = 1/2 → z = ± 1/√2
• 4z ² - 3 = 0 → z ² = 3/4 → z = ± √3/2
Solve for y in each case above:
• z = sin(y °) = 0 → y = 0 + 360n or y = 180 + 360n
(where n is any integer)
These solutions can be combined into one family, y = 180n.
• z = sin(y °) = -1/√2 → y = 225 + 360n or y = 315 + 360n
• z = sin(y °) = 1/√2 → y = 45 + 360n or y = 135 + 360n
These solutions can also be combined, y = 45 + 90n.
• z = sin(y °) = -√3/2 → y = 240 + 360n or y = 300 + 360n
• z = sin(y °) = √3/2 → y = 60 + 360n or y = 120 + 360n
These can be combined as y = 60 + 180n or y = 120 + 180n.
Solve for x in each case:
• y = x - 20 = 180n → x = 20 + 180n
• y = x - 20 = 45 + 90n → x = 65 + 90n
• y = x - 20 = 60 + 180n → x = 80 + 180n
• y = x - 20 = 120 + 180n → x = 140 + 180n