Gegee: P̂ = 119° en Q̂ = 61° Bepaal: 3.1.1 cosec P × tan Q 3.1.2 cos²(P̂ + 2Q) Gegee: rac{1}{2}tan heta = 2, waar heta ext{ e } [0° ; 90°] Toon, sonder die gebruik van 'n sakrekenaar, dat sin² heta + cos² heta = 1 3.3 Los op vir x: sin x = tan 318°, waar x ext{ e } [0° ; 360°] - NSC Technical Mathematics - Question 3 - 2024 - Paper 2

To calculate cos²(P̂ + 2Q):

1. First calculate 2Q:

   $2Q = 2 imes 61° = 122°$

2. Now find P + 2Q:

   $P + 2Q = 119° + 122° = 241°$

3. Calculate cos(241°):

   Using a calculator, we find:

   $ext{cos}(241°) ext{ approximates to } -0.4067$

4. Finally, calculate cos²(241°):

   $ext{cos}^2(241°) = (-0.4067)^2 ext{ which equals approximately } 0.1654$

Starting with the equation:

rac{1}{2}tan heta = 2

1. Multiply both sides by 2:

   $tan heta = 4$

2. Now, find heta:

   $heta = tan^{-1}(4)$

Using a calculator, we find:

$heta ext{ approximates to } 75.96°$

We start with the Pythagorean identity:

$sin² θ + cos² θ = r²$

where $r = 1$ for unit circle representation. This leads to:

1. For sin θ:

   sin² θ = (rac{ ext{opposite}}{ ext{hypotenuse}})^2 = rac{(1)^2}{(1)^2} = 1

2. For cos θ:

   cos² θ = (rac{ ext{adjacent}}{ ext{hypotenuse}})^2 = rac{(0)^2}{(1)^2} = 0

Hence,

$sin² θ + cos² θ = 1 + 0 = 1$

To solve for x:

1. Compute tan 318°:

   $an(318°) = an(-42°) ext{ (since } 318° = 360° - 42°)$

Using a calculator, we find:

$an(318°) ext{ approximates to } -0.9004$

2. Set the equation:

   $sin x = -0.9004$

3. To find x, we look for angles in the 3rd and 4th quadrants:

$x ≈ 243.21° ext{ (3rd quadrant) and } 295.79° ext{ (4th quadrant) }$