Algebra IIUnit: Trigonometric Functions

Chapter 13 – Trigonometric Functions

13.1 Right Triangle Trigonometry

Goals for this lesson•Find Values of Trigonometric Functions for Acute Angles•Solve Problems Involving Right Triangles

This section starts on page 700 in your text.

Solve for the missing sides:(NOT DRAWN TO SCALE; You have 10 minutes!!)

13.1 Vocabulary:1. Trigonometry2. Trigonometric functions3. Sine4. Cosine5. Tangent6. Cosecant7. Secant8. Cotangent9. Solve a right triangle10. Angle of elevation11. Angle of depression

The Americans with Disabilities Act (ADA) provides regulations designed to make public buildings accessible to all.

Under this act, the slope of an entrance ramp designed for those with mobility disabilities must not exceed a ratio of 1 to 12.

This means that for every 12 units of horizontal run, the ramp can rise or fall no more than 1 unit.

*Using this information draw and label the entrance ramp. (only one run ) Answer : What is formed?

When viewed from the side, a ramp forms a right angle.

The slope of the ramp can be described by the tangent of the angle the ramp makes with the ground.

In this example, the tangent of angle A is .12

1

TRIGONOMETRIC VALUESThe tangent of an angle is one of the ratios used in trigonometry.

Trigonometry is the study of the relationships among the angles and sides of a right triangle.

Consider the right triangle in which the measure of an acute angle is identified by the Greek letter theta,

ABCA .

The sides of the triangle are the hypotenuse, the leg opposite and the leg adjacent to

.

Using these sides, you can define six trigonometric functions:sinecosinetangentcotangentsecantcosecant.

These functions are abbreviated:sincostancotseccsc

These trigonometric ratios are functions (of θ) so we can talk about the domain and range of the functions.

The domain of each of these trigonometric functions is the set of all acute angles of a right triangle.

The values of the functions depend only on the measure of and not on the size of the right triangle.

For example, consider in the figure.

Using

Using

sin

:ABCAB

BCsin

:''CAB'

''sin

AB

CB

The right triangles are similar because they share angle

Since they are similar, the ratios of corresponding sides are equal.

That is,

Therefore, you will find the same value for regardless of which triangle you use.

.

'

''

AB

CB

AB

BC

sin

Example 1

Find the values of the six trigonometric functions for angle .

Example 2

Find the values of the six trigonometric functions for angle .

Throughout the trigonometric unit, a capital letter will be used to denote both a vertex of a triangle and the measure of the angle at that vertex.

The same letter in lowercase will be used to denote the side opposite that angle and its measure.

USE ONE TRIGONOMETRIC RATIO TO FIND ANOTHER

If find the value of ,5

2cos A .tan A

Example 3

If find the value of ,3

5tan A .csc A

Angles that measure and occur frequently in trigonometry.

The table below gives the values of the six trigonometric functions for these angles.

To remember these values, use the properties of and triangles.

,45,30 60

906030 904545

RIGHT TRIANGLE PROBLEMSWrite an equation involving sin, cos, or tan that can be used to find the value of x. Then solve the equation. Round to the nearest hundredth.

Example 4Write an equation involving sin, cos, or tan that can be used to find the value of x. Then solve the equation. Round to the nearest hundredth.

If you know the measures of any two sides of a right triangle or the measures of one side and one acute angle, you can determine the measures of all the sides and angles of the triangle.

This process of finding the missing measures is known as solving a right triangle.

SOLVING A RIGHT TRIANGLESolve Round all measures to the nearest hundredth..XYZ

Solve Round all measures to the nearest hundredth..XYZExample 5

To find the measures of angles when one of its trigonometric ratios is known, you will have to use the inverse function on your calculator.

For example, use function to find the measure of an angle when the sine of the angle is known.

Make sure you know how to do this on your calculator. Make sure your calculator is set to degrees.

1sin

?...)5.0(sin5.030sin 1

Solve Round all measures to the nearest hundredth..ABCExample 6

Solve Round all measures to the nearest hundredth..ABCExample 7

Trigonometry has many practical applications.

Among the most important is the ability to find distances or lengths that either cannot be measured directly or are not easily measured directly.

INDIRECT MEASUREMENTIn order to construct a bridge across a river, the width of the river at that location must be determined.Suppose a stake is planted on one side of the river directly across from a second stake on the opposite side.At a distance 50 meters to the left of the stake, an angle of is measured between the two stakes.Find the width of the river.

82

Suppose, in a similar situation the angle was measured at a distance 30 meters away from the stake, and found to be Find the width of the river.

.55

Some applications of trigonometry use an angle of elevation or depression.In the figure the angle formed by the line of sight from the observer and a line parallel to the ground is called the angle of elevation.The angle formed by the line of sight from the plane and a line parallel to the ground is called the angle of depression.

The angle of elevation and the angle of depression are congruent .Why?

The Aerial run in Snowbird, Utah, has an angle of elevation of Its vertical drop is 2900 feet. Estimate the length of this run.

.2.20

Classwork:

Homework: 13.1 Practice