91Ó°ÊÓ

In triangle \(A B C, \angle A=40^{\circ}, a=15,\) and \(b=20\) (a) Show that there are two triangles, \(A B C\) and \(A^{\prime} B^{\prime} C^{\prime},\) that satisfy these conditions. (b) Show that the areas of the triangles in part (a) are proportional to the sines of the angles \(C\) and \(C^{\prime},\) that is, $$\frac{\text { area of } \triangle A B C}{\text { area of } \triangle A^{\prime} B^{\prime} C^{\prime}}=\frac{\sin C}{\sinC^{\prime}}$$

Find the exact value of the trigonometric function. $$\cot \left(-\frac{\pi}{4}\right)$$

Find the exact value of the expression. $$\tan \left(\sin ^{-1} \frac{12}{13}\right)$$

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. $$-\frac{\pi}{4}$$

Find the area of the triangle whose sides have the given lengths. \(a=11, \quad b=100, \quad c=101\)

Show that, given the three angles \(A, B, C\) of a triangle and one side, say \(a\), the area of the triangle is $$\text { area }=\frac{a^{2} \sin B \sin C}{2 \sin A}$$

Find the exact value of the expression. $$\cot \left(\sin ^{-1} \frac{2}{3}\right)$$

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. $$-45^{\circ}$$

Find the exact value of the trigonometric function. $$\cos \frac{7 \pi}{4}$$

Rewrite the expression as an algebraic expression in \(x\). $$\cos \left(\sin ^{-1} x\right)$$