Triangle \(ABC\) is a right triangle with \(C=90^{\circ}\). If \(a=6\) and \(c=10\), find the six trigonometric functions of \(A\).

Fill in the blanks so that each expression becomes a true statement.

1. \(\sin\underline{\qquad}=\cos{30^{\circ}}\)

2. \(\tan{y}=\cot\underline{\qquad}\)

3. \(\sec{75^{\circ}}=\csc{\underline{\qquad}}\)

Show that the following are true.

1. \(\cos^2{30^{\circ}}+\sin^2{30^{\circ}}=1\)

2. \(\cos^2{45^{\circ}}+\sin^2{45^{\circ}}=1\)

Let \(x=30^{\circ}\) and \(y=45^{\circ}\) in each of the expressions that follow, and then simplify each expression as much as possible.

1. \(2\sin x\)

2. \(\sin{2y}\)

3. \(4\sin(3x-90^{\circ})\)

Add \(48^\circ 49'\) and \(72^\circ 26'\).

Subtract \(24^\circ 14'\) from \(90^\circ\).

Change \(27.25^\circ\) to degrees and minutes.

Change \(10^\circ 45'\) to decimal degrees.

Use a calculator to find \(\cos 37.8^\circ\).

Use a calculator to find \(\tan 58.75^\circ\).

Use a calculator to find \(\sec 78^\circ\).

Find the acute angle \(\theta\) for which \(\tan\theta=3.152\). Round your answer to the nearest tenth of a degree.

Find the acute angle \(A\) for which \(\sin A=0.3733\). Round your answer to the nearest tenth of a degree.

To the nearest hundredth of a degree, find the acute angle \(B\) for which \(\sec B=1.0768\).

Find the acute angle \(C\) for which \(\cot C=0.0975\). Round to the nearest degree.

In the right triangle \(ABC\), \(A=40^{\circ}\) and \(c=12\) centimeters. Find \(a\), \(b\), and \(B\).

In the right triangle \(ABC\), \(a=2.73\) and \(b=3.41\). Find the remaining sides and angles.

The two equal sides of an isosceles triangle are each \(24\) centimeters. If each of the two angles measure \(52^{\circ}\), find the length of the base and altitude.

If a \(75.0\)-foot flagpole casts a shadow \(43.0\) feet long, to the nearest \(10\) minutes, what is the angel of elevation of the sun from the tip of the shadow?

A man climbs \(213\) meters up the side of a pyramid and finds that the angle of depression to his starting point is \(52.6^{\circ}\). How high off the ground is he?

A boat travels on a course bearing N \(52^{\circ}\, 40'\) E for a distance of \(238\) miles. How many miles north and how many miles east has the boat traveled?

An arrow is shot into the air so that its horizontal velocity is 25 feet per second and it vertical velocity is 15 feet per second. Find the velocity of the arrow.

The human canonball is shot from a cannon with an initial velocity of \(53\) miles per hour at an angle of \(60^\circ\) from the horizontal. Find the magnitudes of the horizontal and vertical components of the velocity vector.

A boat travels \(72\) miles on a course of bearing N \(27^\circ\) E and then changes its course to travel \(37\) miles at N \(55^\circ\) E. How far north and how far east has the boat traveled on this \(109\)-mile trip?