1) The document introduces the Sine Ratio and how it can be used to find the measure of a side or angle in a right triangle when the lengths of other sides and/or measures of other angles are known.
2) Examples are provided to demonstrate how to set up and solve problems using the Sine Ratio, labeling triangles and identifying the relevant trigonometric ratio formula.
3) New terms for angle of elevation and angle of depression are defined and an example problem involving finding the height of a kite is shown.
Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
For any right triangle
Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
the slides uploaded above are a way to learn and master the basics of trigonometry, the explain and define all three triangles and the hypotenuse rule, which is the Pythagoras.
Trigonometry is a field of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Though many modern students who pursue this career are unaware of its history, it has always been important to engineers. Pythagoras, the Pythagorean Theorem and Archimedes’ The Great Bridge are just a few of the timeless concepts that the mathematics of triangles can be applied to. Trigonometry has two primary components: arithmetic and geometry. Geometry describes geometric relationships between lines and angles. Arithmetic is the study of multiplication, division, integration, and multiplication.
The data is present below the pictures so as to edit it as per your needs. I wanted to use good fonts and this was the only way i could do it as the fonts would not be available on your computer.
For any right triangle
Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
the slides uploaded above are a way to learn and master the basics of trigonometry, the explain and define all three triangles and the hypotenuse rule, which is the Pythagoras.
Trigonometry is a field of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Though many modern students who pursue this career are unaware of its history, it has always been important to engineers. Pythagoras, the Pythagorean Theorem and Archimedes’ The Great Bridge are just a few of the timeless concepts that the mathematics of triangles can be applied to. Trigonometry has two primary components: arithmetic and geometry. Geometry describes geometric relationships between lines and angles. Arithmetic is the study of multiplication, division, integration, and multiplication.
The data is present below the pictures so as to edit it as per your needs. I wanted to use good fonts and this was the only way i could do it as the fonts would not be available on your computer.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
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Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
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1. The Sine Ratio
Slide 1
In the previous lesson, you used the Pythagorean Theorem to find the measure of
a missing side (of a right triangle).
Over the next few lessons, you will learn different methods for finding the measure
of a side or an angle.
How to label triangles:
• (theta) – usually used to identify an angle (a capital letter can also be used to
identify an angle)
• adjacent – the side that is adjacent to the angle
• opposite – the side that is opposite to the angle
• hypotenuse – the side that is directly across from the right angle.
2. Slide 2
When labelling the right triangle, everything depends on which angle
you plan to work with.
Reminder:
• adjacent – the side that is adjacent to the angle
• opposite – the side that is opposite to the angle
• hypotenuse – the side that is directly across from the right angle.
A
B C
A
B C
opposite
opposite
3. Slide 3
When labelling the right triangle, everything depends on which angle
you plan to work with.
Reminder:
• adjacent – the side that is adjacent to the angle
• opposite – the side that is opposite to the angle
• hypotenuse – the side that is directly across from the right angle.
A
B C
A
B C
adjacent
adjacent
4. Slide 4
Tip to help remember which side is which.
Straight line across from angle = opposite (P has a straight line as
part of its shape)
Curved line from angle = adjacent (J is a curved letter)
A
B C
A
B C
adjacent
opposite
opposite
adjacent
5. Slide 5
Calculator Check …
When working with the trigonometric ratios: sine, cosine and tangent,
you must make sure your calculator is set to Degree Mode.
Quick Check:
sin 30 = 0.5
If you do not get this as an answer,
then your calculator is not in
degree mode.
6. Slide 6
The Sine Ratio
Formal definition: in a right triangle, the ratio of the length of the side
opposite a given angle to the length of the hypotenuse
A
B C
sin
opp
hyp
adjacent
opposite
7. Example 1:
PQR is shown below. Using the SINE ratio, find the measure of x
(round to 1 decimal place).
Slide 7
x
67
120
8. Example 1:
PQR is shown below. Using the SINE ratio, find the measure of x
(round to 1 decimal place).
Slide 8
x
67
120
Label the triangle.
Identify the formula.
adjacent
opposite
hypotenuse
sin
opp
hyp
9. Example 1:
PQR is shown below. Using the SINE ratio, find the measure of x
(round to 1 decimal place).
Slide 9
x
67
120
Label the triangle.
Identify the formula.
adjacent
opposite
hypotenuse
sin
opp
hyp
sin 67
120
x
0.9205
120
x
*Recommendation:
Use 4 decimal places
0.9205
1
120 1 0
20
2
x
110.46
110.5
x
x
10. Example 2:
PQR is shown below. Using the SINE ratio, find the measure of x
(round to 1 decimal place).
Slide 10
5
23
x
11. Example 2:
PQR is shown below. Using the SINE ratio, find the measure of x
(round to 1 decimal place).
Slide 11
5
23
x
Label the triangle.
Identify the formula.
sin
opp
hyp
adjacent
opposite
hypotenuse
5
sin 23
x
5
0.3907
x
*Recommendation:
Use 4 decimal places
5
0.3907
x
x x
0.3907 5x
0.3907
0.3907
07
5
0.39
x
12.7975
12.8
x
x
12. Slide 12
New Terms:
Angle of Elevation
Angle of Depression
When you look up at an airplane flying
overhead, the angle between the
horizontal and your line of sight is called
an angle of elevation.
When you look down from a cliff to a
boat passing by, the angle between the
horizontal and your line of sight is called
an angle of depression.
13. Example 3:
The angle of elevation of Sandra’s kite string is 70. If she has
let out 55 feet of string, and is holding the string 6 feet above the
ground, how high is the kite?
Slide 13
14. Example 3:
The angle of elevation of Sandra’s kite string is 70. If she has
let out 55 feet of string, and is holding the string 6 feet above the
ground, how high is the kite?
Slide 14
Label the triangle.
Identify the formula.
sin
opp
hyp
adjacent
oppositehypotenuse
15. Example 3:
The angle of elevation of Sandra’s kite string is 70. If she has
let out 55 feet of string, and is holding the string 6 feet above the
ground, how high is the kite?
Slide 15
Label the triangle.
Identify the formula.
sin
opp
hyp
adjacent
oppositehypotenuse
sin 70
55
h
0.9397
55
h
0.9397
5
55 55
5
h
51.6835
51.7
51.7
57.7
6
h
h
h
h feet
*Held 6 feet
above ground