Trigonometric functions, identities and equations: แบบฝึกหัด

Find the exact value of \(\sin 105^\circ\).

Given that \( \sin \theta = p \) and \( \theta \) is an obtuse angle, express \( \cos 2\theta \) in terms of \( p \).

The function \( f(x) = a \cos(bx) + c \) has a maximum value of 7, a minimum value of -1 and a period of \( \pi \). Find the values of the constants \( a, b \) and \( c \), where \( a \) and \( b \) are positive integers.

Simplify the expression \( \frac{\sin 3\theta}{\sin \theta} - \frac{\cos 3\theta}{\cos \theta} \) for \( \theta \neq \frac{n\pi}{2} \).

Find the maximum and minimum values of the expression \( 5 \sin \theta - 12 \cos \theta + 3 \).

Find the exact value of \(\tan\left(\frac{4\pi}{3}\right) + \cos\left(\frac{7\pi}{6}\right)\).

Find the principal value of \(\cos^{-1}\left(-\frac{1}{2}\right)\) in radians and hence solve \(\cos x = -\frac{1}{2}\) for the interval \(0 \le x \le 2\pi\).

Prove the identity \(\frac{1 - \cos 2\theta}{\sin 2\theta} = \tan \theta\), and hence find the exact value of \(\tan 22.5^\circ\) in the form \(\sqrt{a} - b\) where \(a\) and \(b\) are integers.

Given that \(\sin \theta = -\frac{5}{13}\) and \(180^\circ < \theta < 270^\circ\), find the exact value of each of the following without using a calculator:
(a) \(\cos \theta\),
(b) \(\tan 2\theta\).

(a) Prove the identity \(\frac{\sin 2A}{1 + \cos 2A} = \tan A\).
(b) Hence, solve the equation \(\frac{\sin 2A}{1 + \cos 2A} = 2 \sin A\) for \(0^\circ \le A \le 360^\circ\).