Gegee: sin β = rac{1}{3} , waar β ∈ (90°; 270°) Sonder die gebruik van 'n sakrekenaar, bepaal elk van die volgende: 5.1.1 cos β 5.1.2 sin 2β 5.1.3 cos(450° - β) 5.2 Gegee: rac{cos^2 x + sin^2 x imes cos^2 x}{1 + sin x} 5.2.1 Bewys dat rac{cos^2 x + sin^2 x imes cos^2 x}{1 + sin x} = 1 - sin x 5.2.2 Vir wat ter waardes(s) van x in die interval x ∈ [0°; 360°) is rac{cos^2 x + sin^2 x imes cos^2 x}{1 + sin x} ongedefinieerd? 5.2.3 Skryf die minimum waarde neer van die funksie gedefinieer deur g = rac{cos^2 x + sin^2 x imes cos^2 x}{1 + sin x} Gegee: cos(A - B) = cosA cosB + sinA sinB 5.3.1 Gebruik die identiteit hiervo vor om af te lei dat sin(A - B) = sinA cosB - cosA sinB 5.3.2 Bepaal vervolgens andersins die algemene oplossing van die vergelyking sin 48° cos x - cos 48° sin x = cos 2x 5.4 Vereenvoudig sin 3x + sin x cos 2x + 1 - NSC Mathematics - Question 5 - 2023 - Paper 2

Using the double angle formula for sine:
[ sin 2β = 2 \cdot sin β \cdot cos β
]
Substituting for sin β and cos β:
[ sin 2β = 2 \cdot \frac{1}{3} \cdot -\frac{2\sqrt{2}}{3} = -\frac{4\sqrt{2}}{9}
]

Using the cosine angle difference formula:
[ cos(450° - β) = cos 450° \cdot cos β + sin 450° \cdot sin β
]
Since ( cos 450° = cos(360° + 90°) = 0 ) and ( sin 450° = sin 90° = 1 ), we have:
[ cos(450° - β) = 0 \cdot cos β + 1 \cdot sin β = sin β = \frac{1}{3}
]

Starting with the left-hand side:
[ LHS = \frac{cos^2 x + sin^2 x \cdot cos^2 x}{1 + sin x}
]
Factor out cos^2 x:
[ = \frac{cos^2 x (1 + sin^2 x)}{1 + sin x}
]
Noting that ( 1 + sin^2 x = (1 - sin x)(1 + sin x) ):
[ = \frac{cos^2 x (1 - sin x)(1 + sin x)}{1 + sin x}
]
Thus, cancelling ( (1 + sin x) ):
[ = cos^2 x (1 - sin x)
]
Since ( cos^2 x + sin^2 x = 1 ), we simplify:
[ LHS = 1 - sin x
]
Thus, LHS = RHS.

The expression is undefined when the denominator equals zero:
[ 1 + sin x = 0 \Rightarrow sin x = -1
]
This occurs at:
x = 270°.
Thus, the expression is undefined at ( x = 270° ).

To find the minimum value of g, evaluate the function's behavior:
From our earlier analysis, we note that
[ g = cos^2 x (1 - sin x)
]
The function will achieve its minimum when ( sin x = 1 ) which gives us the minimum value of:
[ g_{min} = 0 \quad (as , sin x \to 1)
]

Using the provided identity:
[ cos(A - B) = cos A cos B + sin A sin B
]
We can write:
[ sin(A - B) = sin A cos B - cos A sin B
]
This follows from the sine-cosine angle relationship.

Using the sine difference formula:
[ sin(48° - x) = cos 2x
]
Equating both sides for the general solution:
[ 48° - x = 2kπ \quad or \quad 48° - x = -2kπ \pm 90°
]
Solving for x gives us:
x = 48° - 2kπ;
[ x = 48° - 2kπ + 90° \text{ for k in } ℤ ]

Using addition for trigonometric functions:
[ sin 3x + sin x = 2 sin \left( \frac{3x + x}{2} \right) \cos \left( \frac{3x - x}{2} \right)
]
This simplifies to:
[ = 2 sin(2x) cos(x)
]
Additionally, we simplify the denominator further to find the final expression.