Product of trig functions

Solve the equation \(\cos x + \sqrt{(1 - \frac{1}{2} \sin 2x)} = 0\), in the interval \(0° \leq x < 360°\). [9]

Jessica, a maths student, is asked by her teacher to solve the equation \(\tan x = \sin x\), giving all solutions in the range \(0° \leq x \leq 360°\) The steps of Jessica's working are shown below. \(\tan x = \sin x\) Step 1 \(\Rightarrow\) \(\frac{\sin x}{\cos x} = \sin x\) Write \(\tan x\) as \(\frac{\sin x}{\cos x}\) Step 2 \(\Rightarrow\) \(\sin x = \sin x \cos x\) Multiply by \(\cos x\) Step 3 \(\Rightarrow\) \(1 = \cos x\) Cancel \(\sin x\) \(\Rightarrow\) \(x = 0°\) or \(360°\) The teacher tells Jessica that she has not found all the solutions because of a mistake. Explain why Jessica's method is not correct. [2 marks]

Solve the equation $$\sin\theta\tan\theta + 2\sin\theta = 3\cos\theta \quad \text{where } \cos\theta \neq 0$$ Give all values of \(\theta\) to the nearest degree in the interval \(0° < \theta < 180°\) Fully justify your answer. [5 marks]

Solve the following equation for values of \(\theta\) between \(0°\) and \(360°\). $$3\tan\theta + 2\cos\theta = 0$$ [6]

Solve \(2 \sin x = \tan x\) exactly, where \(-\frac{\pi}{2} < x < \frac{\pi}{2}\). [4]

Solve the equation $$\sin\theta\tan\theta + 2\sin\theta = 3\cos\theta \quad \text{where } \cos\theta \neq 0$$ Give all values of \(\theta\) to the nearest degree in the interval \(0° < \theta < 180°\) Fully justify your answer. [5 marks]

Solve the following trigonometric equation in the range given. $$4\tan^2\theta\cos\theta = 15, \quad 0 \leq \theta < 360°.$$ [6 marks]