Right Triangles and Trigonometry

Finding Trigonometric Ratios

A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The word trigonometry is derived from the ancient Greek language and means measurement of triangles. The three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated as sin, cos and tan, respectively.

Finding Trigonometric Ratios

Example 2: Find the sine, the cosine, and the tangent of the indicated angle.

a. Ð S b. Ð R

Solution: a. The length of the hypotenuse is 13. For Ð S, the length of the opposite side is 5, and the length of the adjacent side is 12.

You can find trigonometric ratios for 30 ° , 45 ° and 60 ° by applying what you know about special right triangles.

Trigonometric Ratios for 45 °

Example 3: Find the sine, the cosine, and the tangent of 45 ° .

Solution: Begin by sketching a 45 ° – 45 ° – 90 ° triangle. Because all such triangles are similar, you can make calculations simple by choosing 1 as the length of each leg. From Theorem 9.8 on page 551, it follows that the length of hypotenuse is .

» 0.7071

» 0.7071

» 1

USING TRIGONOMETRIC RATIOS IN REAL LIFE

The angle that your line of sight makes with a line drawn horizontally is called the angle of elevation.

Estimating a Distance

Example 7: Escalators: The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles rises 76 feet at a 30 ° angle. To find the distance d a person travels on the escalator stairs, you can write a trigonometric ratio that involves the hypotenuse and the known leg lehgth of 76 feet.

Write ratio for sine of 30 ° .

Substitute.

d sin 30 ° = 76 Multiply each side by d .

Divide each side by sin 30 °

Substitute 0.5 for sin30 ° .

Simplify.

A person travels 152 feet on the escalator stairs.

Trigonometric functions

sine( q ) = opp/hyp cosecant( q ) = hyp/opp

cosine( q ) = adj/hyp secant( q ) = hyp/adj

tangent( q ) = opp/adj cotangent( q ) = adj/opp

The functions are usually abbreviated: sine (sin), cosine (cos), tangent (tan) cosecant (csc), secant (sec), and cotangent (cot).

It is often simpler to memorize the trigonometric functions in terms of only sine and cosine:

The functions are usually abbreviated:

arcsine (arcsin)
arccosine (arccos)
arctangent (arctan)
arccosecant (arccsc)
arcsecant (arcsec)
arccotangent (arccot).

According to the standard notation for inverse functions (f -1 ), you will also often see these written as sin -1 , cos -1 , tan -1 arccsc -1 , arcsec -1 , and arccot -1 . Beware , though, there is another common notation that writes the square of the trig functions, such as (sin(x)) 2 as sin 2 (x). This can be confusing, for you then might then be lead to think that sin -1 (x) = (sin(x)) -1 , which is not true. The negative one superscript here is a special notation that denotes inverse functions (not multiplicative inverses).

Trigonometry : Labeling Sides

Since there are three sides and two non-right angles in a right triangle, the trigonometric functions will need a way of specifying which sides are related to which angle. (It is not-so-useful to know that the ratio of the lengths of two sides equals 2 if we do not know which of the three sides we are talking about. Likewise, if we determine that one of the angles is 40°, it would be nice to know of which angle this statement is true.

We need a way of labeling the sides. Consider a general right triangle:

A right triangle has two non-right angles, and we choose one of these angles to be our angle of interest, which we label " q ." (" q " is the Greek letter "theta.")

We can then uniquely label the three sides of the right triangle relative to our choice of q . As the above picture illustrates, our choice of q affects how the three sides get labeled.

We label the three sides in this manner: The side opposite the right angle is called the hypotenuse . This side is labeled the same regardless of our choice of q . The labeling of the remaining two sides depend on our choice of theta; we therefore speak of these other two sides as being adjacent to the angle q or opposite to the angle q . The remaining side that touches the angle q is considered to be the side adjacent to q , and the remaining side that is far away from the angle q is considered to be opposite to the angle q , as shown in the picture.

Measurement of Angles

The word trigonometry comes from two Greek words, trigonon and metron , meaning “triangle measurement”. The earliest use of trigonometry may have been for surveying land in ancient Egypt after the Nile River 's annual flooding washed away property boundaries.

In trigonometry, an angle often represents a rotation about a point. Thus, the angle q shown is the result of rotating its initial ray to terminal ray .

A common unit for measuring very large angles is the revolution, a complete circular motion. For example, when a car with wheels of radius 14 in. is driven at 35 mi/h, the wheels turn at an approximate rate of 420 revolutions per minute (abbreviated rpm).

A common unit for measuring smaller angles is the degree, of which there are 360 in one revolution. For example, when a door is opened, the doorknob is usually turned revolution, or 90 degrees.

Angles can be measured more precisely by dividing 1 degree into 60 minutes, and by dividing 1 minute into 60 seconds. For example, an angle of 25 degrees, 20 minutes, and 6 seconds is written 25 ° 20 ¢ 66 ¢ ¢ .

Angles can also be measured in decimal degrees. To convert between decimal degrees and degrees, minutes, and seconds, you can reason as follows:

12.3 ° = 12 ° + 0.3(60) ¢ = 12 ° 18 ¢

25 ° 20 ¢ 6 ¢ ¢ = 25 ° + = 25.335 °

Example: a. Convert 196 ° to radians (to the nearest hundredth).

b. Convert 1.35 radian to decimal degrees (to the nearest tenth) and to degrees and minutes (to the nearest ten minutes).

Solution: Use a calculator and the conversion formulas above. Note that some calculators have the conversion formulas already built in; consult the instruction manual for your calculator.

a. 196 ° = 4196 ´ » 3.42 radians

b. 1.35 radians = 1.35 ´ » 77.3 ° » 77 ° 20 ¢

Angle measures that can be expressed evenly in degrees cannot be expressed evenly in radians, and vice versa. That is why angles measured in radians are frequently given as fractional multiples of p . Angles whose measures are multiples of , and appear often in trigonometry. The diagrams below will help you keep the degree conversions for these special angles in mind. Note that a degree measure, such as 45 ° , is usually written with the degree symbol ( ° ), while a radian measure, such as , is usually written without any symbol.

	Multiples of
	Multiples of
	Other multiples of

When an angle is shown in a coordinate plane, it usually appears in standard position, with its vertex at the origin and its initial ray along the positive x–axis. Moreover, we consider a counterclockwise rotation to be positive and a clockwise rotation to be negative. The diagrams below give examples of positive and negative angles. (In this book we often do not distinguish between an angle and its measure. Thus, by “ positive and negative angles” we mean angles with positive and negative measures.)

If the terminal ray of an angle in standard position lies in the first quadrant, as shown at the left above, the angle is said to be a first–quadrant angle. Second–, third–, and fourth–quadrant angles are similarly defined. If the terminal ray of an angle in standard position lies along an axis, as shown at the right above, the angle is called a quadrantal angle. The measure of a quadrantal angle is always a multiple of 90 ° , or .

Two angles in standard position are called coterminal angles if they have the same terminal ray. For any given angle there are infinitely many coterminal angles.

Example: Find two angles, one positive and one negative, that are coterminal with the angle . Sketch all three angles.

Solution: A positive angle coterminal with is:

+ 2 p =

A negative angle coterminal with is:

– 2 p = –

The three angles are shown below.

Equipment

To get started, all you need is a computer with high-speed Internet connection.

Tell-a-Friend