Section 6.2 Trigonometry of Right Triangles – p.443 This lesson begins our study of Trigonometric Ratios and Functions. The most common ratios are sine, cosine, and tangent. But these ratios are also used as functions in many applications. For example, Many digital compression algorithms, like JPEG and MP3 use fourier transforms that rely on sine and cosine functions. Surveying, navigation and astronomy all rely on sine and cosine for the position of objects and other calculations. Music is composed of waves of different frequencies and amplitudes and these can be described using sine and/or cosine. In fact most anything involving sound waves is usually described in terms of a sine or cosine function. Ballistic trajectories rely on sine and/or cosine, as well as other applications in physics. Space flight relies on calculations and conversions to polar coordinates using sine and cosine. So do satellites. GPS and cellphones rely on triangulation and formulas involving sine and/or cosine. Signal transmission, like for TV and radio broadcasting, involves waves described with sine and/or cosine graphs.

Thermal analysis is used to model how things get hot (electronics, spacecraft, ovens, etc). This equation is usually solved using sums of sines and cosines. Signal Processing - The whole area of digital signal processing, which is used for HDTV and digital audio, is based on using sums of sines and cosines. Geology - Earthquakes are modeled using using sums of sines and cosines. Building Design - Buildings need to be designed to resist wind and earthquakes. The effect of waves on buildings is often modeled using sines and cosines to simulate wind and earth motion. These simulations determine how the buildings oscillate, which is also modeled by sines and cosines. We will eventually be seeing some of these uses in the examples we examine. But first, we need to learn the basics of trigonometric functions and that starts by examining trigonometric ratios. You probably remember trigonometry from your studies in Geometry. Here is a definition of trigonometry, trigonometry - the branch of mathematics that deals with the relationships between the sides and the angles of triangles and the calculations based on them, particularly the trigonometric functions

This means any kind of triangle, but we will start with right triangles in this section since those are more useful than other kinds of triangles in your later studies. Below, we see 2 different right triangles with our angle of interest labeled using the greek capital “T” called theta or .

You may remember from your Geometry studies the mnemonic device “SOHCAHTOA” which helps you to remember the sine, cosine, and tangent ratios. But, there are 3 more the reciprocals of those 3 ratios - cosecant(reciprocal of sine), secant(reciprocal of cosine), and cotangent(reciprocal of tangent). When we list the ratios, we usually abbreviate the word opposite as “opp”, adjacent as “adj”, and hypotenuse as “hyp”. From now on, we will simply refer to them as functions because eventually we will be using them in their functional sense.

Definition of Trigonometric Functions - Let be an acute angle of a right triangle. The six trigonometric functions of

are defined as follows:

or

In order to evaluate these functions for an angle, we need to know all 3 side lengths of the right triangle. Example: Find the value of all six trigonometric functions for the given right triangle,

From the Pythagorean Theorem, we see the length of the missing side is 24. So, our six functions are as follows:

sin opp

hypcos

adj

hyptan

opp

adj

csc hyp

oppsec

hyp

adjcot

adj

opp

csc 1

sinsec

1

coscot

1

tan

sin 7

25cos

24

25tan

7

24

csc 25

7sec

25

24cot

24

7

There were also some other special right triangles due to their side lengths being integers(or Pythagorean triples). Some of the more common ones are: 3-4-5 5-12-13 8-15-17 7-24-25 Remember that ANY multiple of these special right triangles are also right triangles such as:

6-8-10 15-36-39 12-22.5-25.5 7 -24 -25 You may also recall that we have special right triangles by the angles in the right triangle. Those are:

and . Let’s take a closer look at these triangles. The side lengths of these triangles relate in the following way:

*You really should know how the side lengths relate to each other and the values for all six trigonometric functions in these two special right triangles. Those values are:

5 5 5

30o 60o 90o 45o 45o 90o

Do you notice any relationship between sine, cosine, and

tangent? --> . This is another relationship you should know. We can use some of these values to find the missing parts of a right triangle, i.e. “solve” the right triangle. Example: Solve the right triangles using the given angle.

sin30o 1

2cos30o

3

2tan30o

3

3

csc30o 2sec30o

2 3

3 cot 30o 3

sin60o 3

2cos60o

1

2 tan60o 3

csc60o 2 3

3 sec60o 2cot60o

3

3

sin45o 2

2cos45o

2

2 tan45o 1

csc45o 2 sec45o 2 cot 45o 1

tan sin

cos

Of course we can use our calculator to solve right triangles. Note(Important): Make sure your calculator is in degree mode. Press the MODE key on your calculator and highlight DEGREE. 1.) In a waterskiing competition, a jump ramp has the measurements shown on the diagram below. What is the height, h, above water at which the skier leaves the ramp?

2.) A skateboard ramp has a height of 12 inches and the angle

between the ramp and the ground is 17 . Find the length, d, of the base of the ramp.

sin opp

hyp sin15.1o

h

19 h 4.95 feet

Now, you try the following examples: A.) Solve the triangle -->

B.) A kite is flown 4 feet above the ground at an angle of 48 degrees from a horizontal line where the person holding the kite is looking straight ahead. If the kite line is 500 feet long, how high above the ground is the kite?

tan opp

adj tan17o

12

d d 39.25 feet

tan19o a

13, cos19o

13

c

B 71o, a 4.48, c 13.8

When you add the 4 feet above the ground where the kite is held, the kite would be 376 feet above the ground. Another common type of problem using trig. functions is “angle of elevation” and “angle of depression”. Both of these angles are measured relative to a horizontal line coming straight out from the eyes of the observer. See the diagram below.

sin48o d

500

d 372 feet

In the kite example above, we could have said that the angle that the kite makes with the horizontal line made by the eyes of the observer is called the angle of elevation of the kite. Example: A biologist whose eye level is 6 ft above the ground

measures the angle of elevation to the top of a tree to be 38.7 . If the biologist is standing 180 ft. from the tree’s base, what is the height of the tree to the nearest foot? *Make sure to always draw a diagram for these types of problems.

Example: A surveyor whose eye level is 6 ft above the ground

measures the angle of elevation to the top of a hill to be 60.7 . If the surveyor is standing 200 feet from the bottom of the hill, what is the straight line distance from the surveyor to the top of the hill?

Now, you try the following examples:

tan38.7o h

180

h 144 feet

h 6 150 feet

cos60.7o 200

hyp

hyp 409 feet

A.) An airplane flying at an altitude of 30,000 feet is headed toward an airport. To guide the airplane to a safe landing, the airplane receives radar signals from the control tower at an

angle of depression of 10 . If the control tower is 50 feet above the ground, what is the horizontal distance from the airplane to the control tower?

B.) A soccer player kicks a soccer ball in a straight line at an

angle of elevation of 40 with the ground. If the ball leaves the ground at a speed of 5 m/s, what is the height of the ball after one minute?

*Hmwk: From the problems below, do 5-17 every other odd, 23-39 every other odd, 45, 47, 50, 51

sin40o h

300

h 192.8 feet

tan80o d

29,950

d 169,855 feet