(Solved): Question 6. Find the two values of \( \theta \) in the interval \( -180^{\circ} ...

Question 6. Find the two values of \( \theta \) in the interval \( -180^{\circ}<\theta \leq 180^{\circ} \) such that \[ \cos (\theta)=-0.70 \] Enter your answers to \( \theta_{1} \) and \( \theta_{2} \) in degrees, correct to two decimal places. [4 Marks] Question 7. A wave of the form \( y=3 \sin (6 t)+2 \cos (6 t) \quad \) can be expressed in the form \[ y=R \cos (6 t-\alpha) \quad \text { Where }-\pi \leq \alpha \leq \pi \] If time \( t \) is measured in seconds, determine: (a) The amplitude of oscillation, (b) Phase angle (in radians), (c) The angular frequency, and (d) The fundamental period of the resulting waveform. Give all your answers correct to 2 decimal places. [8 Marks] Question 8. An exponentially growing quantity \( P \) is modelled by \( P=16 e^{3 k t} \) If \( P=3154 \) at \( t=1.57 \), then what is the value of \( k \) to three decimal places? [4 Marks] Given the complex number \( z_{1}=3+4 i \) and \( z_{2}=5-4 i \). Determine \( \frac{z_{1}}{z_{2}} \) in the form of \( a+i b \) and find the real and imaginary parts of \( \frac{z_{1}}{z_{2}} \) correct to three decimal places. [6 Marks] Question 10. Given the exponential form of complex numbers \[ \omega_{1}=1.8 e^{2.1 i} \text { and } \omega_{2}=-2.9 e^{0.2 i} \] Determine the modulus, \( r \), and argument, \( \theta \), of \[ z=\frac{\left(\omega_{1}\right)^{3}}{\left(\omega_{2}\right)^{2}} \quad \text { Where }-\pi<\theta<\pi . \] [6 Marks]