What you should learn

To use trigonometric ratios to solve rigth triangles

NCTM Curriculm Standards 2, 4, 6 - 10

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Trigonometric Ratios

Hypotenuse

Legs

Angle of elevation

Angle of depression

Introduction: The Leaning Tower of Pisa, a bell tower in Pisa, Italy, currently leans 16.5 feet off center. The tower was closed in 1990 in order for workers to attempt to stabilize the tower's foundation to prevent it from eventually collapsing. By 1994, the tower's lean had decreased about 2/5 of an inch.

If you were to draw a line from the top of the tower to the ground, a right triangle would be formed with the ground. If enough is known about a right triangle, certain ratios can be used to find the measures of the remaining parts of the triangle. These ratios are called trigonometric ratios.

A typical right triangle is shown below.

a is the measure of the side opposite A, BC.

b is the measure of the side opposite B, AC.

c is the measure of the side opposite C, AB.

a is adjacent to B and C.

b is adjacent to A and C.

c is adjacent to A and B.

The side opposite C, the right angle, is called the hypotenuse. The other two sides are called legs.

Three common trigonometric rations are defined as follows.

 

Definition of Trigonometric Ratios:

sine of A = measure of leg opposite A/measure of hypotenuse

sin A = a/c

cosine of A = measure of leg adjacent to A/measure of hypontenuse

cos A = b/c

tangent of A = measure of leg opposite to A/measure of leg adjacent to A

tan A = a/b

 

Notice that sine, cosine, and tangent are abbreviated as sin, cos, and tan, respectively.

Exercise 1: Find the sine, cosine, and tangent of each acute angle. Round your answers to the nearest thosandth.

sin J = opposite leg/hypotenuse = 7/25 = 0.208

cos J = adjacent leg/hypotenuse = 24/25 = 0.960

tan J = opposite leg/adjacent leg = 7/24 = 0.292

Now you do sin, cos, and tan of angle L...

 

 

Consider the triangles QTV and MSO. Since corresponding angles have the same measures, QTV ~MSO. Recall that in similar triangles, the corresponding sides are proportional.

m/q = s/t

(q/s) * (m/q) = (q/s) * (s/t)

m/s = q/t

Sin M = Sin Q

In general, the sine of a 64 degree angle of a right triangle will be the same number no matter how large or small the triangle is. A similar result holds for cosine and tangent.

You can use a calculator to find the values of trigonometric functions or to find the measure of an angle.

Exercise 2: Find the value of sin 64 degrees to the nearest ten thousandth.

Use a scientific calculator

ENTER: 64 0.898794046

Rounded to the nearest ten thousandth, sin 64 = 0.8988

Exercise 3: Find the measure of P to the nearest degree.

Many real-world applications involve trigonometry.

Exercise 4: Refer to the application at the beginning of the lesson. Engineers working to correct the lean of the Leaning Tower of Pisa could check the angle the Tower makes with the ground to find any increase in the lean. Use the measurements to find the angle to the nearest degree that the Tower made witht he ground.

Since the length of adjacent side and hypotenuse are known, use the cosine.

cos x = adjacent leg/hypotenuse

cos x = 16.5/179

cos x = 0.09217877

x = 85 degrees

To the nearest degree, the measure of the angle is 85 degrees

 

 

You can find the missing measures of a right triangle if you know the measure of two sides of the triangle or the measure of one side and one acute angle. Finding all of the measures of the sides and the angles in a right traingle is called solving the triangle.

Exercise 5: Solve ABC.

In the real world, trigonometric ratios are often used to find distances or lengths that cannot be measured directly. In these applications, you will sometimes use an angle of elevation or an angle of depression. An angle of elevation is formed by a horizontal line of sight and a line of sight above it. An angle of depression is formed by a horizontal line of sight and a line of sight below it.

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 21 - 57 odd, 58 - 69

Alternative Homework: Enriched: 22 - 54 even, 55 - 69

Extra Practice: Students book page 765 Lesson 4-3

Extra Practice Worksheet: Click Here.

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