Math: Area, Volume, Circles and Trigonometry

This is Unit Fourteen, part two, of ACT Preparation. Part one closed by choosing sine, cosine or tangent; this part finds angles from sides, extends trigonometry to every angle, and solves triangles with no right angle. The question carries over: is the answer a length, an area, a volume or an angle? Here a second rule joins it: measure every angle from the horizontal. By the end you will be able to find an angle from two sides and solve line-of-sight problems. To use reciprocal functions and radians. To find exact values on the unit circle. To use the Pythagorean identity. And to choose between the laws of sines and cosines.

Three entries on finding angles, and on angles of elevation and depression.

An angle from two sides.

ACT 1320 Solving Right Triangles

To find an angle from two sides, use an inverse trig function: if the sine of A is one half, A is the inverse sine of one half, thirty degrees. Legs seven and seven root three give a tangent of one over root three, so the angles are thirty and sixty. Inverse sine gives an angle; it is not one over sine.

Looking up and looking down.

ACT 1321 Angles of Elevation and Depression

An angle of elevation is measured up from the horizontal to a line of sight; an angle of depression is measured down from it. Looking up at twenty-five degrees to a tower 100 meters away, the height is 100 times the tangent of twenty-five, about 46.6 meters. Both are measured from the horizontal.

A trigonometry word problem, step by step.

ACT 1322 A Right-Triangle Trig Problem

Sketch the right triangle, place the given angle from the horizontal, label the known and wanted sides, choose the ratio, and check the size. From an eighty-meter cliff, a boat at a twenty-degree angle of depression is seen; by alternate interior angles the angle at the boat is twenty, so the distance is eighty over the tangent of twenty, about 220 meters. Placing twenty degrees at the top of the cliff would give about twenty-nine meters, which is wrong.

Six entries on reciprocal functions, radians, and the unit circle.

The reciprocal functions.

ACT 1323 Reciprocal Functions

The reciprocal trig functions are cosecant, one over sine; secant, one over cosine; and cotangent, one over tangent. If the sine is three fifths, the cosecant is five thirds. Pair them by their partners, not their prefixes: cosecant goes with sine.

Radians.

ACT 1324 Radians

A radian is the central angle whose arc equals the radius. A full circle is 2 pi radians, so 180 degrees is pi radians: ninety degrees is pi over two, and three pi over four is 135 degrees. Convert by multiplying: degrees times pi over 180 gives radians.

The unit circle.

ACT 1325 The Unit Circle

On the unit circle, of radius one at the origin, an angle theta from the positive x-axis meets the circle at the point cosine theta, sine theta. At ninety degrees, the point is zero, one. Signs follow the quadrant: in the second quadrant, cosine is negative and sine positive.

Exact values, step by step.

ACT 1326 Exact Values on the Unit Circle

Find the reference angle to the x-axis, take its sine and cosine from the special triangles with hypotenuse one, and attach the quadrant's signs. The sine of thirty is one half, the cosine root three over two, and the sine and cosine of forty-five are root two over two. For 150 degrees, the reference angle is thirty, so the cosine is negative root three over two; for 225, both are negative root two over two. The reference angle is measured to the x-axis: for 150, it is thirty, not sixty.

The Pythagorean identity.

ACT 1327 The Pythagorean Identity

For every angle, sine squared plus cosine squared equals one, because the point lies on the unit circle. If the sine is 0.6, the cosine squared is 0.64, so the cosine is plus or minus 0.8. The quadrant decides the sign, and sine squared means the sine, squared.

Using the identity, step by step.

ACT 1328 Using the Pythagorean Identity

Use the identity to find the missing sine or cosine, choose its sign from the quadrant, then form the tangent and any reciprocal. With cosine negative five thirteenths in the second quadrant, the sine squared is 144 over 169, and the second quadrant makes the sine positive twelve thirteenths; the tangent is negative twelve fifths, the secant negative thirteen fifths. A triangle sketch gives the sizes but not the signs; only the quadrant supplies those.

Four entries on waves, on any triangle, and the whole unit in one structure.

Amplitude and period.

ACT 1329 Amplitude and Period

For a times the sine or cosine of b x, the amplitude is the size of a, half the distance from lowest to highest point, and the period is 2 pi over b. Three times the sine of 2x rises to three, falls to negative three, and repeats every pi. A larger b makes the cycles shorter.

Two laws for any triangle.

ACT 1330 The Laws of Sines and Cosines

In any triangle, each side over the sine of its opposite angle is the same ratio: the law of sines. And c squared equals a squared plus b squared minus 2ab times the cosine of C: the law of cosines. Sides five and eight around sixty degrees give a third side of seven. At ninety degrees, the law of cosines becomes the Pythagorean theorem.

Which law to use, step by step.

ACT 1331 Which Law to Use

List what is known; if an angle and its opposite side are known, plus one more part, use the law of sines; if two sides and the angle between, or all three sides, use the law of cosines. With angles of forty and seventy-five and a side of ten opposite the forty, the law of sines gives about fifteen. Sides seven, eight and thirteen give a cosine of negative one half for the largest angle, so 120 degrees. Find the largest angle first with the law of cosines, because it might be obtuse.

The whole unit in one structure.

ACT 1332 A Silo: Cylinder Plus Hemisphere

Measuring figures comes down to a few formulas and the judgment to apply them. A silo is a cylinder of radius four meters and height fifteen, topped by a hemisphere. The cylinder holds 240 pi and the hemisphere 128 thirds pi, about 888 cubic meters in all. Painting the outside, with the base on the ground, covers the side, 120 pi, and the dome, 32 pi: 152 pi, about 478 square meters. From thirty meters away, the top of the dome, nineteen meters up, is seen at an angle whose tangent is nineteen thirtieths, about thirty-two degrees. Decide whether the answer is a length, an area or a volume; read what is exposed or removed; and place every angle inside its triangle, from the horizontal.

Look at what you can now do. You can find an angle with an inverse function and solve line-of-sight problems from the horizontal. You can pair each reciprocal function with its partner and convert between degrees and radians. You can find exact values on the unit circle, measuring the reference angle to the x-axis. You can find one trig value from another, letting the quadrant choose the sign. And you can read a wave's amplitude and period and choose the right law for any triangle. This ends the geometry; unit fifteen turns to data, chance and counting.