Answer:
see explanation
Step-by-step explanation:
(a)
using the trigonometric identity
• sec x = [tex]\frac{1}{cosx}[/tex], then
7cos x - 3 = 2cos²x ( arrange in standard form )
2cos²x - 7cosx + 3 = 0 ← in standard form
we require the factors of the product 2 × 3 = 6 which sum to - 7
these are - 6 and - 1
use these factors to split the middle term
2cos²x - 6cosx - cosx + 3 = 0 ( factor by grouping )
2cosx(cosx - 3) - 1(cosx - 3) = 0 ( factor out (cosx - 3) )
(cosx - 3)(2cosx - 1) = 0
equate each factor to zero and solve for x
cosx - 3 = 0 ⇒ cosx = 3 ← has no solution
2cosx - 1 = 0 ⇒ cosx = [tex]\frac{1}{2}[/tex] ⇒ x = 60° , x = 300°
solution x = 60°, x = 300° for 0° < x < 360°
(b)
using the trigonometric identity
• cos2x = 1 - 2sin²x, then
1 - 2sin²x - 3sinx - 1 = 0 ← in standard form
- 2sin²x - 3sinx = 0 ( multiply through by - 1 )
2sin²x + 3sinx = 0 ( factor out sinx )
sinx(2sinx + 3) = 0
equate each factor to zero and solve for x
2sinx + 3 = 0 ⇒ sin x = - [tex]\frac{3}{2}[/tex] ← has no solution
sinx = 0 ⇒ x = 180°
solution x = 180° for 0° < x < 360°
