The addition formulae and trigonometric identities are used to simplify or evaluate trigonometric expressions. Trigonometric equations are solved using a double angle formulae and the wave function.

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 \(\alpha\)?

\[k = \sqrt 2 ;\,\alpha = 45^\circ\]

\[k = \sqrt 2 ;\,\alpha = 135^\circ\]

\[k = 2;\,\alpha = 45^\circ\]