Ms. Lim Geometry

Aim: How do we use find angle measures using trigonometry? Learning Target: I can apply trigonometry to find angle measures Learning Objective(s): Students develop an understanding of how to determin a missing angle in a right tirangle diagram and apply to to real-world situations Common Core Learning Standard (CCLS): G.SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Vocabulary: adjacent leg, opposite leg, hypotenuse, leg, right angle, ratio, tangent, sine, cosine, inverse, cofunctions, angle of elevation, angle of depression Video: Khan Academy Strategies: SOLVE

DN: In a right triangle where one of the angles measures 30º, what is the ratio of the length of the side opposite the 30º angle to the length of the side opposite the 90º angle?

1)

2)

3)

4)

Lesson

We have been using trigonometry to find measure of a side of a right triangle.

We can use trig ratios to find the measures of angles in a right D . Verify that your calculator is in

degree mode before using the trig functions.

In , sin A = 5

3. What is the angle that when we take the angle of

it, gives 5

3?

The notation, the inverse sine, sin -1

5

3 represents the angle whose

sine is 5

3.

a. Find the measure of using sin -1

5

3.

b. Use sin -1 notation to represent the angle whose

sine is 5

4.

Inverse Notation:

sin-1

(x) = arcsin(x)

cos-1

(x) = arccos(x)

tan-1

(x) = arctan(x)

arcsin(x) is read "the angle

whose sine is x".

Find the measure of the unknown angle to the nearest degree.

1. 2.

3.

4.

Real Life Application The diagram shows a ramp connecting theground to a loading platform 4.5 feet above the ground. The ramp measures 11.75 feet from the ground to the top of the loading platform.

Determine and state, to the nearest degree, the angle of elevation formed by the ramp and the ground.

Let's return to the second question. What angle will Dan find if the tree is 50 meters tall?

c. Use a calculator to find each angle measure in degrees.

1tan- is read inverse tangent of x.

1- is a symbol for inverse. It is not an exponent.

arcsin

5

3= sin -1 ÷

ø

öçè

æ

5

3

arcsin

5

4= sin -1

5

4

Find the angle to the nearest tenth.

In DABC, CÐ is a right angle, a = 4, and c = 10. Find m AÐ . (23.6)

In DABC, C is a right angle, b = 5, and c = 9. Find m A . (56.3)

The acute angles of a right triangle are complementary. Sine and Cosine are cofunctions.

Find values for that make each statement true.

a.

b.

c.

d.

Worksheet

TIER II appear, determine, expand, explain, justify, prove, relate, summarize, show, identify

MP2: Reason abstractly & quantitatively MP4: Model with mathematics

MP6: Attend to precision.

Materials: Promethean Board, Active Inspire Software, Worksheet, Rubrics

UDL Highlight vocabulary words on projection board Use graphing calculators Essential Questions: How do we express the trigonometric ratios for a given angle? How are the trigonometric ratios related in similar triangles? How can you use the sine and cosine ratios, and their inverses, in calculations involving right triangles? Strategies: Think-Pair-Share, Turn & Talk, Thumbs Up & Down

Information Technology HS Geometry 2 Ms. Lim

Name: ________________________________________________ Period: __________

Angle of Elevation and Angle of Depression

1. Find to the nearest degree the measure of the angle of elevation of the sun when a vertical pole 6

meters high casts a shadow 8 meters long.

2. Find to the nearest foot the height of a tree that casts a 24-foot shadow when the angle of

elevation of the sun is 52˚.

3. From an angle of depression of °, John watches his friend approach his building while standing

on the rooftop. The rooftop is from the ground, and John’s eye level is at about . from

the rooftop. What is the distance between John’s friend and the building?

At a point on the ground 40 meters from the foot of a tree, the measure of the angle of elevation

of the top of the tree is 42 ˚. Find the height of the tree to the nearest meter.

Aim: How do we find angle measures using trigonometry? HW #6

Midterm - Tuesday, April 4

Class Exam - Thursday, April 6

Do Now: Find the values of a

and b to the nearest hundredth.

HW #5 Review

Aim: How do we find angle measures using trigonometry?

HW #7 Midterm - Tuesday, April 4 Class Exam - Thursday, April 6

DN: Given with =8.5 and

=55°, find

and to the nearest hundredth.

HW #5 Review

You can use trig ratios to find the measures of angles in a right

D .

sin A = 5

3

The notation sin -1

÷ø

öçè

æ

5

3 represents the

angle whose sine is 5

3.

Read as “inverse sine of A”.

Express to the nearest degree.

a. Find A = sin -1

÷ø

öçè

æ

5

3 using your calculator.

b. Use sin -1

notation to represent the angle whose sine is 5

4.

arcsin

5

3= sin

-1÷ø

öçè

æ

5

3

arcsin

5

4= sin

-1

5

4

1tan- is read inverse tangent of x. 1-

is a symbol for inverse.

It is not an exponent.

Worksheet

Aim: How do we find model and solve word problems using trigonometry?

Midterm - Tuesday, April 4 Class Exam - Thursday, April 6 HW #8

DN: Find the angle to the nearest tenth.

In DABC, CÐ is a right angle, a = 4, and c = 10. Find m AÐ .

In DABC, C is a right angle, b = 5, and c = 9. Find m A .

Trigonometric Ratios of Complementary Angles

If A and B are acute angles in a right , then

)90cos(sin and

)90sin(cos

Find values for that make each statement true.

sin = cos (25)

sin 80 = cos

sin = )10cos(

)45sin( = cos

Aim: How do we find model and solve word problems using trigonometry?

Castlelearning Project – due on Thurs, April 20 at 11:59pm

DN:

A 28-foot ladder is leaning against a house. The bottom of the

ladder is 6 feet from the base of the house. Find the measure of

the angle formed by the ladder and the ground, to the nearest

degree.

Worksheet

DN: Find x to the nearest thousandth.

Aim: How do we find angle measures using trigonometry?

DN: Find the measure of

angle D. Show your work.

An angle of elevation is an angle

formed by rays that are parts of a

horizontal line and a line of sight that is

elevated, or raised upward, from the

horizontal.

An angle of depression is an angle

formed by rays that are parts of a

horizontal line and a line of sight that is

depressed, or lowered, from the

horizontal.

Student Activity Worksheet

In rhombus ABCD, diagonals AC

and BD meet at M. If BD = 14 and

AC = 20, find to the nearest

degree:

a. BCMm b. MBCm

c. ABCm d. BCDm

Find to the nearest degree the measure of the angle of

elevation of the sun when a vertical pole 6 meters high casts a

shadow 8 meters long.

Find to the nearest foot the height of a tree that casts a 24-foot

shadow when the angle of elevation of the sun is 52˚.

At a point on the ground 40 meters from the foot of a tree, the

measure of the angle of elevation of the top of the tree is 42 ˚.

Find the height of the tree to the nearest meter.

If an engineer wants to design a highway to connect New York

City directly to Buffalo, at what angle, x, would she need to build

the highway? Find the angle to the nearest degree.

Trigonometric Ratios

sine of

hypotenuse

legoppositeA

cosine of

hypotenuse

legadjacentA

tangent of

legadjacent

legoppositeA

The formulas can be remembered by SOH CAH TOA.

Information Technology HS Geometry 4 Ms. Lim

Name: ___________________________________________ Date: ________________

HW #10 – Right Triangles and Trigonometric Ratios

Find the value of x to the nearest tenth.

1.

2.

3.

4.

5.

6.

Information Technology HS Geometry 4 Ms. Lim

Name: ______________ KEY ___________________ Date: ________________

HW #10 – Right Triangles and Trigonometric Ratios

Find the value of x to the nearest tenth.

1. 10.1

2. 12.7

3. 25.4

4. 20.0

5. 3.8

6. 40.2

7.

8.

scratch

Select an angle measure. Then have each member of your group draw a right triangle, ABCwhere A has the selected measure. Make the triangles different sizes.

Measure the legs of each ABC to the nearest millimeter.

1. Compute the ratio Atoadjacentleg

Aoppositeleg

and round to two decimal places.

2. Compare the ratios in your group. Make a conjecture.

Use a calculator to find each angle measure in degrees.

3tan 1- 2

2sin 1-

2

3cos 1-

Scratch

Day 3 Word Problems

HMH p 684

DN: Find x to the nearest thousandth.

(49.502)

Angle of elevation and depression

An angle of elevation is an angle formed by rays that are

parts of a horizontal line and a line of sight that is elevated,

or raised upward, from the horizontal.

An angle of depression is an angle formed by rays that are parts of a horizontal line and a line

of sight that is depressed, or lowered, from the horizontal.

Day 4 How do we solve word problems using trigonometry?

In rhombus ABCD, diagonals AC and BD meet

at M. If BD = 14 and AC = 20, find to the nearest

degree:

a. BCMm b. MBCm

c. ABCm d. BCDm

Find to the nearest degree the measure of the angle of elevation of the sun when a vertical pole

6 meters high casts a shadow 8 meters long (37)

Find to the nearest foot the height of a tree that casts a 24-foot shadow when the angle of

elevation of the sun is 52˚.

At a point on the ground 40 meters from the foot of a tree, the measure of the angle of elevation

of the top of the tree is 42 ˚.. Find the height of the tree to the nearest meter. (36 m)

Geometry CC Regents Trig questions

Jume 2015/ 5, 12, 28, 34

Jan 2016/ 9, 8, 15, 22

3tan 1-

2

2sin 1-

2

3cos 1-

In ABC , mC = 90, AC = 6, and AB = 10. Find tan A.

In RST , mT = 90, RS = 13 and ST = 12, Find tan S.