What is the exact value of \(\sin 75^\circ\)?
\[\frac{{\sqrt 3 - 1}}{{2\sqrt 2 }}\]
\[\frac{{1- \sqrt 3 }}{{2\sqrt 2 }}\]
\[\frac{{1 + \sqrt 3}}{{2\sqrt 2 }}\]
What is the exact value of \(\cos105^\circ\)?
\[\frac{{1 - \sqrt 3 }}{{2\sqrt 2 }}\]
Given that \(a\) is an acute angle and that \(\tan a= \frac{1}{5}\), what is the exact value of \(\sin 2a\)?
\[\frac{1}{{13}}\]
\[\frac{5}{{12}}\]
\[\frac {5}{{13}}\]
When \(\sin (x + 2y)\cos (y - x) + \cos (x + 2y)\sin (y - x)\) is simplified, what does it equal?
\[\cos (2x + y)\]
\[\cos 3y\]
\[\sin 3y\]
When \(\cos 7x^\circ \cos 5x^\circ + \sin 7x^\circ \sin 5x^\circ\) is simplified, what is the answer?
\[\cos 2x^\circ\]
\[\cos 12x^\circ\]
\[\sin 2x^\circ\]
If \(\sin x = \frac{2}{3}\) and \(0 \textless x \textless \frac{\pi }{2}\), find the exact values of \(\cos x\) and \(\tan x\).
\(\cos x = \frac{{\sqrt 5 }}{3}\) and \(\tan x = \frac{2}{{\sqrt 5 }}\)
\(\cos x = \frac{{\sqrt 5 }}{9}\) and \(\tan x = \frac{4}{{\sqrt 5 }}\)
\(\cos x = \frac{2}{{\sqrt 5 }}\) and \(\tan x = \pm \frac{{\sqrt 5 }}{3}\)
When \(\cos x + \sqrt 3 \sin x\) is expressed in the form \(k\sin (x + \alpha )\), what are the values of \(k\) and \(\alpha\)?
\(k = 2\) and \(\alpha = 30^\circ\)
\(k = 2\) and \(\alpha = 60^\circ\)
\(k = \sqrt 2\) and \(\alpha = 30^\circ\)
The maximum value of \(5\sin (x - 30)^\circ\) occurs where \(x\) equals:
\[120^\circ\]
\[90^\circ\]
\[60^\circ\]
What is the maximum value of \(2 + 3\sin (x + 20)^\circ\)?
\[3\]
\[- 1\]
\[5\]
Given that \(k\cos \alpha = - 1\) and \(k\sin \alpha = 1\), \(k \textgreater 0;\,0 \le \alpha \le 360\), what are the values of \(k\) and \(\alpha\)?
\[k = \sqrt 2 ;\,\alpha = 45^\circ\]
\[k = \sqrt 2 ;\,\alpha = 135^\circ\]
\[k = 2;\,\alpha = 45^\circ\]