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Unit 6

Introduction to Trigonometry

Lesson 1: Incredibly Useful Ratios

Opening Exercise

Use right triangle [pic] to answer 1–3.

1. Name the side of the triangle opposite [pic] in two different ways.

2. Name the side of the triangle opposite [pic] in two different ways.

3. Name the side of the triangle opposite [pic] in two different ways.

For each triangle, label the appropriate sides as hypotenuse, opposite, and adjacent with respect to the marked acute angle.

|4. |5. |

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Vocabulary

o The side of a right triangle opposite the right angle

is called the _____________________________________.

o The side of a right triangle opposite the marked

acute angle is called the __________________________ side.

o The 3rd side of a right triangle (one of the two rays

of the marked acute angle) is called the ____________________________ side.

Exploratory Challenge

For each triangle in your set, label the sides with respect to the given angle as hyp, opp, and adj. Then, set up the [pic] and [pic] ratios and place them in the table.

|Group 1 |

| |Triangle |Angle Measures |Side Lengths |[pic] |[pic] |

|1. |[pic] |[pic] |[pic] [pic] | | |

|2. |[pic] |[pic] |[pic] | | |

|3. |[pic] |[pic] |[pic] | | |

|4. |[pic] |[pic] |[pic] | | |

|5. |[pic] |[pic] |[pic] | | |

|Group 2 |

| |Triangle |Angle Measures |Length Measures |[pic] |[pic] |

|1. |[pic] |[pic] |[pic] | | |

|2. |[pic] |[pic] |[pic] | | |

|3. |[pic] |[pic] |[pic] | | |

|4. |[pic] |[pic] |[pic] | | |

|5. |[pic] |[pic] |[pic] | | |

Reduce each ratio using your calculator, and round to the nearest hundredth.

With a partner from the other group of triangles, fill in the other table.

What you can conclude about each pair of triangles between the two sets?

Examples

Refer back to your completed chart from the Exploratory Challenge and record the decimal value that we calculated for the ratios of the sides. Using these ratios, for each question solve for the missing side lengths and round to the nearest tenth if necessary.

|class example: |[pic] |[pic] |

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|[pic] | | |

|1. |[pic] |[pic] |

|[pic] | | |

|2. |[pic] |[pic] |

|[pic] | | |

3. From a point 120 meters away from a building, Serena measures the angle between the ground and the top of a building and finds that it measures [pic]. What is the height of the building? Round to the nearest meter.

Homework

1. Indicate which leg is adjacent and which is opposite the given angle of [pic].

2. Using the value of the ratio [pic], find the approximate length of the

hypotenuse rounded to the nearest tenth.

3. Using the Pythagorean Theorem, find the length of the 3rd side of the given right triangle.

4. Rationalize the denominator of [pic].

Lesson 2: The Definition of Sine, Cosine, and Tangent

Opening Exercise

1. Label the sides of each triangle with respect to the circled angle as hyp, opp, and adj.

2. Identify the [pic] ratios for [pic] and [pic].

3. Identify the [pic] ratios for [pic] and [pic].

4. Describe the relationship between the ratios for angles [pic] and [pic]

There is a name for each of these incredibly useful ratios that we have looked at. These trigonometric ratios were discovered long ago and have had several names. The names currently in use are translations of Latin words. Mathematicians have agreed upon the

names __________________, _________________________, and ___________________________.

In trigonometry, we sometimes represent the measure of the angle with the Greek letter [pic], pronounced “theta”.

If [pic] is the angle measure of [pic] as shown, then we define:

The sine of [pic] is the value of the ratio of the length of the opposite side to the length of the hypotenuse.

As a formula, [pic] In the given diagram, [pic]

The cosine of [pic] is the value of the ratio of the length of the adjacent side to the length of the hypotenuse.

As a formula, [pic] In the given diagram, [pic]

The tangent of [pic] is the value of the ratio of the length of the opposite side to the length of the adjacent side.

As a formula, [pic] In the given diagram, [pic]

Examples

1. Using [pic], complete the following table.

(Do not simplify the ratios.)

|[pic] name |[pic] measure |[pic] |[pic] |[pic] |

|[pic] | | | | |

|[pic] | | | | |

2. Using [pic], complete the following table.

(Do not simplify the ratios.)

|[pic] name |[pic] measure |[pic] |[pic] |[pic] |

|[pic] | | | | |

|[pic] | | | | |

Important Discovery!

Describe the patterns that you notice from the charts.

Exercises

1. Tamer did not finish completing the table below for a diagram similar to the previous problems that we just completed. In the diagram, [pic] represents the measure of [pic] and [pic]represents the measure of [pic]. Use any patterns you notice from Exercises 1 and 2 to complete the table for Tamer.

|Measure of Angle |Sine |Cosine |Tangent |

|[pic] |[pic] |[pic] |[pic] |

|[pic] | | | |

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2.

a. Using the given diagrams, find the ratios for [pic] and [pic].

b. Reduce these ratios. What do you notice?

c. Would this also work for cosine and tangent?

d. Why or why not?

Homework

1. a. Find the length of [pic].

b. Using the information from [pic], complete the table.

(Do not simplify the ratios.)

|Angle Measure |[pic] ratio |[pic] ratio |[pic] ratio |

|[pic] | | | |

|[pic] | | | |

2. If [pic] and [pic] are the measures of complementary angles such that [pic] and [pic] , label the sides and angles of the right triangle in the given diagram with possible side lengths.

3. Given [pic] and [pic] in right triangle ABC, complete the missing values in the table. You may draw a diagram to help you. (Do not simplify the ratios.)

|Angle Measure |[pic] |[pic] |[pic] |

|A |[pic] |[pic] |[pic] |

|B | | | |

Lesson 3: Sine and Cosine of Complementary and Special Angles

Opening Exercise

There are certain special angles where it is possible to give the exact value of sine and cosine. These frequently seen angles are [pic], [pic], [pic], [pic], and [pic].

[pic]

Using the given triangles, complete the following table and rationalize the denominators if necessary.

|[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |

|Sine |0 | | | |[pic] |

|Cosine |1 | | | |[pic] |

Find two values in the table that are the same. What do you notice about the angle measures?

Find a different set of values in the table that are the same. What do you notice about their angle measures?

In the opening exercise we noticed that when angles [pic] and [pic] are ________________________________, [pic].

A rule to remember is:

Examples

1. Consider the right triangle [pic] where [pic] is a right angle.

a. Find the sum of [pic].

b. What can you conclude about [pic] and [pic]?

c. What can you conclude about [pic] and [pic]?

2. Find the values for [pic] that make each statement true.

a. [pic]

b. [pic]

c. [pic]

d. [pic]

|Ratio of Sides of Special Right Triangles |

|30 – 60 – 90 triangle |45 – 45 – 90 triangle |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

Exercises

1. Find the missing side lengths in the given triangle.

2. Find the missing side lengths in the given triangle.

Homework

1. Find the missing side lengths in the given triangle.

2. [pic] is a 30 – 60 – 90 triangle.

Find the unknown lengths for [pic] and [pic].

3. Find the values for [pic] that make each statement true.

|a. [pic] |b. [pic] |c. [pic] |

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4. Given an equilateral triangle with sides of length 9, find the length of the altitude. Confirm your answer with the use of the Pythagorean Theorem.

Lesson 4: Solving Problems Using Sine and Cosine

Opening Exercise

Based on Lesson 3, make a prediction about how the sum of [pic] will relate to the sum of [pic].

Use the sine and cosine values of special angles to find the sum of[pic].

Use the sine and cosine values of special angles to find the sum of [pic].

Was your prediction a valid prediction? Explain why or why not.

We have been working with trigonometric ratios for sine, cosine, and tangent. We will now use our calculators to find the values of [pic], [pic], and [pic]. Your graphing calculator must be in DEGREE MODE to be able to calculate the trig values since they are all degree measures. Radians are a measure that we will use in a later unit and next year in Algebra II.

EXAMPLES

You will need a calculator

|[pic] |[pic] |

4. An anchor cable supports a vertical utility pole forming a [pic] angle with the ground. The cable is attached to the top of the pole. If the distance from the base of the pole to the base of the cable is 5 meters, how tall is the pole rounded to the nearest hundredth?

5. Determine the measure of the indicated angle to the nearest degree.

6. A pendulum consists of a spherical weight suspended at the end of a string whose other end is anchored at a pivot point [pic]. The distance from [pic] to the center of the pendulum’s sphere, [pic], is 6 inches. The weight is held so that the string is taught at a horizontal, as shown in the diagram, and then dropped.

a. Calculate the angle of rotation, to the nearest degree, when the pendulum drops 2 inches from the horizontal.

b. Calculate the angle of rotation, to the nearest degree, when the pendulum drops 4 inches from the horizontal.

Homework

Doug is installing a surveillance camera inside a convenience store. He mounts the camera [pic] above the ground and [pic] horizontally from the store’s entrance. The camera is meant to monitor every customer that enters and exits the store.

a. If the average customer is between [pic] and [pic] feet tall, what is a reasonable distance from the ceiling to the sight line of the camera?

b. Set up a trig ratio using your answer from part a to find the angle of depression from the camera.

c. Calculate the angle of depression to the nearest degree.

d. If the camera were moved closer to the door, how would this affect the angle of depression?

Lesson 9: Trigonometry and the Pythagorean Theorem

Opening Exercise

1. In a right triangle, with acute angle of measure [pic], [pic]. What is the value of [pic]?

2. In a right triangle, with acute angle of measure [pic], [pic]. Using the same process as #1, what is the value of [pic] in simplest radical form?

Using the marked diagram, we will now discover a trigonometric identity from our knowledge of sine and cosine.

[pic] [pic]

Using the trigonometric ratios, rewrite the leg lengths

in terms of [pic] and [pic] on the blank triangle.

What if we did the Pythagorean Theorem? It states that for a right triangle with lengths [pic], where [pic] is the hypotenuse, the relationship between the side lengths is [pic]. Let’s apply the Pythagorean Theorem to this triangle.

Important Discovery!

This new Pythagorean Theorem: [pic]

is called the ___________________________________ _______________________________.

This identity is sometimes written in a more concise way: _________________________________________.

Example 1

You will need a calculator

In a right triangle, with acute angle of measure [pic], [pic]. Use the Pythagorean Identity to determine the value of [pic].

Example 2

You will need a calculator

In a right triangle, with acute angle of measure [pic], [pic]. Use the Pythagorean Identity to determine the value of [pic].

Another trigonometric identity can be found using the relationship between sine, cosine, and tangent. Now, if we let [pic], what do the ratio values become?

[pic]

|In Example 1, we found that when [pic], then[pic]. Using these values |In Example 2, we found that when [pic], then [pic]. Using |

|and the new trigonometric identity for tangent, what is the ratio for |these values and the new trigonometric identity for tangent, what |

|[pic]? |is the ratio for [pic]? |

| |(be sure to rationalize the denominator) |

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If one of the values of [pic] is given, we can find the other two using the identities [pic] and [pic].

Examples

1. If [pic] , use trigonometric identities to find [pic] and [pic].

2. If [pic], find [pic] and [pic].

Homework

1. If [pic], use trigonometric identities to find [pic] and [pic].

2. If [pic], use trigonometric identities to find [pic] and [pic].

3. Using trigonometric ratios, find the height of the pyramid to the nearest tenth.

-----------------------

from table

x

16

20

[pic]

D

O

G

[pic]

iff is shorthand for if and only if

a

6

b

9

8.5

[pic]

[pic]

[pic]

trig values that [pic]

To solve [pic] in your calculator:

o check that your mode is in DEGREES

o turn the equation into [pic]

o which is the same thing as [pic]

o press [pic] and your calculator will display [pic]

o type in [pic] as [pic] using the division key

o hit enter to see the angle measure that has a sine value of [pic]

Be sure to show this work on your paper!

To solve for angle measures in a right triangle using trigonometry:

o draw a diagram if one is not provided

o label given sides of the triangle using hyp, opp, and adj

o determine which trigonometric ratio to use

o set up the trig ratio

o check that your calculator mode is in DEGREES

o using your calculator, solve for the angle measure

28

40

[pic] means approximately

Camera

Steps to solve:

1- draw diagram and label the angle and known side lengths

2- find the length of the 3rd side

3- find the exact value for [pic] (leave as a ratio)

With radicals, we get a [pic] answer, but since these are side lengths, we only use the [pic] answer

1

a

b

It is always true that:

[pic]

[pic]

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