Warm-Up 1 Find the value of x.. Warm-Up 1 Find the value of x

Warm-Up 1Warm-Up 1

Find the value of x.

Warm-Up 1Warm-Up 1

Find the value of x.

History LessonHistory Lesson

Right triangle trigonometry is the study of the relationship between the sides and angles of right triangles. These relationships can be used to make indirect measurements like those using similar triangles.

History LessonHistory Lesson

Early mathematicians discovered trig by measuring the ratios of the sides of different right triangles. They noticed that when the ratio of the shorter leg to the longer leg was close to a specific number, then the angle opposite the shorter leg was close to a specific number.

Example 1Example 1

In every right triangle in which the ratio of the shorter leg to the longer leg is 3/5, the angle opposite the shorter leg measures close to 31. What is a good approximation for x?

Example 2Example 2

In every right triangle in which the ratio of the shorter leg to the longer leg is 9/10, the angle opposite the shorter leg measures close to 42. What is a good approximation for y?

Trig RatiosTrig Ratios

The previous examples worked because the triangles were similar since the angles were congruent. This means that the ratios of the sides are equal.

In those cases we were using the tangent ratio. Here’s a list of the three you’ll have to know.

sine cosine tangent

Trigonometric Ratios ITrigonometric Ratios I

Objectives:

1. To discover the three main trigonometric ratios

2. To use trig ratios to find the lengths of sides of right triangles

Investigation 1Investigation 1

Use the GSP Activity to discover the three main Trigonometric ratios sine, cosine, and tangent.

SummarySummary

AB

C

hypotenuse

hypotenuse

side adjacent side adjacent ΘΘ

side o

pp

osite

side o

pp

osite ΘΘ

hypotenuseoppositesin

hypotenuseadjacentcos

adjacentoppositetan

tan OfAlgebra

SummarySummary

cos AnotherHour

sin OhHell

AB

C

hypotenuse

hypotenuse

side adjacent side adjacent ΘΘ

side o

pp

osite

side o

pp

osite ΘΘ

SohCahToaSohCahToa

hypotenuseoppositesin

hypotenuseadjacentcos

adjacentoppositetan

Soh

Cah

Toa

Example 3Example 3

Find the values of the six trig ratios for α and β.

Activity: Trig TableActivity: Trig Table

On the previous example, we knew all the sides of the triangle, and we just listed the three trig ratios for those sides using a generic angle. Usually, though, you know the angle, and you want to find a side.

Nowadays, we would use a calculator to find the sine or tangent of an angle. In the long, dark years before the calculator, people had to find their trig ratios in a table.


In the 1500s, Georg Rheticus, a student of Copernicus, was the first to define the six trig functions in terms of right triangles. He was also the first to start a book of values for these ratios, accurate to ten decimal places to be used in astronomical calculations.


Step 3: Set up a table of values like so:

θ sin θ cos θ tan θ

20°

70°


Step 4: Now use your calculator to round each calculation to the nearest thousandths place.

θ sin θ cos θ tan θ

20°

70°


Step 5: Finally, let’s check your values with those from the calculator.

For sin, cos, and tan

1. Make sure your calculator is set to DEGREE in the MODE menu.

2. Use one of the 3 trig keys. Get in the habit of closing the parenthesis.

Example 4Example 4

To the nearest meter, find the height of a right triangle if one acute angle measures 35° and the adjacent side measures 24 m.

Example 5Example 5

To the nearest foot, find the length of the hypotenuse of a right triangle if one of the acute angles measures 20° and the opposite side measures 410 feet.

Example 6Example 6

Use a special right triangle to find the exact values of sin(45°) and cos(45°).

Example 8Example 8

Find the value of x to the nearest tenth.1. x = 2. x = 3. x =

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Warm-Up 1 Find the value of x.. Warm-Up 1 Find the value of x