Exact trigonometric values：練習問題

Evaluate the exact value of the trigonometric expression:
\(2 \cos 60^{\circ} + \tan 45^{\circ}\)

Evaluate the exact value of $$\sin 60^\circ \times \cos 30^\circ + \tan 45^\circ$$.

In triangle \(ABC\), \(\angle ABC = 90^\circ\), \(\angle BAC = 30^\circ\) and the hypotenuse \(AC = 12\). Point \(D\) lies on the side \(AB\) such that \(\angle BCD = 45^\circ\).

Find the exact length of \(AD\).

Find the exact value of the expression:

\(\frac{4 \sin 30^\circ \tan 60^\circ}{\cos 45^\circ}\)

Evaluate the exact value of the expression:
\(2 \cos 60^{\circ} + \tan 45^{\circ}\)

This question is about using exact trigonometric values.

(a) Evaluate the exact value of the expression:
\( \sin^2 60^\circ + \cos^2 60^\circ \)

(b) Find the exact value of the following expression, giving your answer as a single fraction:
\( \frac{\tan 30^\circ \times \sin 60^\circ}{\cos 0^\circ} \)

In triangle \(ABC\), the length of side \(AB = 8\sqrt{2}\) cm, angle \(ABC = 45^\circ\) and angle \(BCA = 30^\circ\). Point \(D\) lies on the line segment \(BC\) such that \(AD\) is perpendicular to \(BC\).

(a) Find the exact length of \(AD\) by using the exact value of \(\sin 45^\circ\).
(b) Calculate the exact length of side \(AC\).
(c) Show that the exact length of the side \(BC\) can be written as \(8 + 8\sqrt{3}\) cm.
(d) Find the exact area of triangle \(ABC\), giving your answer in the form \(p + p\sqrt{q}\) where \(p\) and \(q\) are integers.