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.