CHAPTER 3 – Right Triangle Trigonometry



CHAPTER 3 – Right Triangle Trigonometry

Trigonometry is the branch of mathematics that studies the relationships between angles and the lines that form them in triangles. It was first developed for use in astronomy and geography. Today, trigonometry is used in surveying, navigation, engineering, construction, and the sciences to explore the relationships between the side lengths and angles of triangles.

Lesson 1

The Tangent Ratio

A trigonometric ratio is a ratio of the measures of two sides of a right triangle.

One trigonometric ratio is the tangent ratio.

The short form for the tangent ratio of angle A is tan A.

Example 1:

Write each trigonometric ratio.

a) tan A

b) tan B

Example 2:

a) Calculate tan 25( to four decimal places.

b) Draw a triangle to represent tan( = 5/4. Calculate the angle ( to the nearest tenth of a degree.

Example 3: Find x.

Example 4: In (PQR, (R = 90(, (P = 27(, and QR = 5 cm. Calculate the length of PR to the nearest tenth of a centimeter.

Example 5: A ladder leaning against a wall forms an angle of 63( with the ground. How far up the wall will the ladder reach if the foot of the ladder is 2 m from the wall?

Example 6: Wes is flying a kite at an angle of 57(, 1 m above ground. The kite and Wes are 8.5 m apart along the ground.

a) At what height from the ground is the kite the flying?

b) How long of a string is being pulled?

c) What angle of elevation is Wes standing from the ground?

Assignment Pg. 108 #7-9, 11-13 + 6.3-Tangent worksheet

Tangent Quiz on _________________________

Lesson 2

The Sine and Cosine Ratio

In the previous lesson, you learned about the tangent ratio. There are two other trigonometric ratios that compare the lengths of the sides of a right triangle. These ratios, called the sine and cosine ratio, involve the hypotenuse.

Example 1:

Write each trigonometric ratio.

a) sin A c) cos A

b) sin B d) cos B

Example 2:

a) Evaluate each ratio, to four decimal places: (i) sin 42( (ii) cos 68(

b) Determine each angle measure, to the nearest degree. (i)sin ( = 0.4771 (ii)cos ( = 0.7225

Example 3: Find the missing side or angle.

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Example 4: A 6.1 m ladder leans against a wall. The angle formed by the ladder and the ground is 71(.

a) How far is the base of the ladder from the wall?

b) How far up the wall does the ladder reach?

Example 5: A skateboarder jumped off a 5 m ramp. The ramp was 3 m above ground. What was the angle of the ramp with the ground?

Assignment Pg. 121-123 #7-14 + 6.4-6.5 worksheet

Sine and Cosine Quiz on ____________________

Lesson 3

Solving Right Triangles

Solve the triangles shown. Express each measurement to the nearest whole unit.

Angles of elevation and depression: The line of sight is the invisible line from one person or object to another person or object. Some applications of trigonometry involve an angle of elevation and an angle of depression.

• An angle of elevation is the angle formed by the horizontal and a line of sight above the horizontal.

• An angle of depression refers to the angle formed by the horizontal and a line of sight below the horizontal.

• How do you think the angle of elevation and the angle of depression are related in the following diagram?

Example 1: A building is 50 feet high. At a distance away from the building, an observer notices that the angle of elevation to the top of the building is 41º. How far is the observer from the base of the building?

Example 2: An airplane is flying at a height of 2 miles above the ground. The distance along the ground from the airplane to the airport is 5 miles. What is the angle of depression from the airplane to the airport?

Example 3: A bird sits on top of a lamppost. The angle of depression from the bird to the feet of an observer standing away from the lamppost is 35˚. The distance from the bird to the observer is 25 meters. How tall is the lamppost?

Example 4: Two poles on horizontal ground are 60 m apart. The shorter pole is 3 m high. The angle of depression of the top of the shorter pole from the top of the longer pole is 20˚. How tall is the longer pole?

Example 5: A man who is 2 m tall stands on horizontal ground 30 m from a tree. The angle of elevation of the top of the tree from his eyes is 28˚. Determine the height of the tree.

Assignment: Pg 131-135 #6-10, 12-15 + 6.6 worksheet

Practice Test on __________________ Test on _______________________

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12 m

A way to remember the different trigonometric ratios is by the acronym SOH(CAH(TOA

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