Basic Trigonometry

1
SOHCAHTOA

• passport to Trigonometry Land ...

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2
Objectives

• CCSS.Math.Content.HSG-SRT.C.6 Understand that by
  similarity, side ratios in right triangles are
  properties of the angles in the triangle, leading
  to definitions of trigonometric ratios for acute
  angles.
• CCSS.Math.Content.HSG-SRT.C.7 Explain and use the
  relationship between the sine and cosine of
  complementary angles.
• CCSS.Math.Content.HSG-SRT.C.8 Use trigonometric
  ratios and the Pythagorean Theorem to solve right
  triangles in applied problems.

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3
Oops! Not a triangle!

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• I am sorry.
• I am not a triangle!
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I am a Triangle!

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• Good job!
• You know me!
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What is a triangle?

• A triangle is a polygon made up of three
  connected line segments in such a way that each
  side is connected to the other two.

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Examples of triangles...

• All these polygons are tri-gons and commonly
  called triangles

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These are not triangles...

• None of these is a triangle...

Can you tell why not?
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Try it yourself...

• Click on each one that IS a triangle?

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Right triangles

• A right triangle is a special triangle that has
  one of its angles a right angle.
• You can tell it is a right triangle when when one
  angle measures 900 or the right angle is marked
  by a little square on the angle whose measure is
  900.

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These are right triangles...
All right triangles
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These are not right triangles...

• These triangles are NOT right triangles. Explain
  why not?

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Try it yourself ...

• Click on the triangle that is NOT a right
  triangle?

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13
I am a right triangle

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• I am sorry you did not recognize me as one.
• I am a right triangle!

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Correct That is not a right Triangle

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• Well done!
• You know your right triangles well!
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Hypotenuse

• The longest side of a right triangle is the
  hypotenuse.
• The hypotenuse lies directly opposite the right
  angle.
• The legs may be equal in length or one may be
  longer than the other.

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Parts of a right triangle ...

• A right triangle has two legs and a hypotenuse...

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Try it yourself hypotenuse

• Click on the side that is the hypotenuse of the
  right triangle.

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Try it yourself shorter leg

• Click on the side that is the shorter leg of the
  right triangle.

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Try it yourself longer leg

• Click on the side that is the longer leg of the
  right triangle.

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Correct Way to go!

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• Excellent!
• Wise choice.
• You know your parts!
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21
Incorrect Choice!

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• I am sorry.
• You chose the wrong side!
• CliCk here to Review Try Again!

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Pythagorean Theorem...

• The right triangle has a special property, called
  the Pythagorean Theorem, that can help us find
  one side if we know the other two sides.

If the lengths of hypotenuse and legs are c, a
and b respectively, then c2 a2 b2
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Finding a side - hypotenuse

• Use the Pythagorean Theorem to find the length of
  the missing side.

c2 a2 b2 x2 102 142 100 196
296 x sqrt(296) 17.2
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Finding a side leg

• Use the Pythagorean Theorem to find the length of
  the missing side.

c2 a2 b2 152 102 x2 225 100
x2 x2 125 x sqrt(125) 11.18
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Try it yourself- hypotenuse

• Find the hypotenuse of the given right triangle
  with the lengths of the legs known

Click on the selection that matches your answer
A. 36
B. 10
C. 100
D. 64
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Try it yourself leg

• Find the leg of the given right triangle with the
  lengths of the leg and hypotenuse known

Click on the selection that matches your answer
A. 24
B. 6
C. 144
D. 12
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Correct Answer

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• You take after Pythagoras!
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Oops not quite!

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• I am sorry.
• Check your calculations again.
• Click to Review Try Again!

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Opposite or Adjacent side?

• In a right triangle, a given leg is called the
  adjacent side or the opposite side, depending on
  the reference acute angle.

Adjacent or opposite from an acute reference
angle refers only to legs and not the hypotenuse
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Opposite side (to an acute angle)

• In a right triangle, a given leg is called the
  adjacent side or the opposite side, depending on
  the reference acute angle.

leg2 is opposite to acute angle A leg1 is NOT
opposite to acute angle A
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Adjacent side (to an acute angle)

• In a right triangle, a given leg is called the
  adjacent side or the opposite side, depending on
  the reference acute angle.

leg1 is adjacent to acute angle A leg2 is NOT
adjacent to acute angle A
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Try it yourself opposite

• Click on the side that is opposite to angle B.

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Try it yourself adjacent

• Click on the side that is adjacent to angle B.

B
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Correct reference!

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• You understood these references!
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35
Not exactly

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• I am sorry.
• Opposite is across from the angle.
• Adjacent is next to the angle.
• Hypotenuse is neither adjacent nor opposite.
• Click to Review Try Again!

36
Trigonometric ratios of an acute angles of a
right triangle

• The ratios of the sides of a right triangle have
  special names.
• There are three basic ones we will consider
• sine
• cosine
• tangent

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37
The sine of an acute angle...

• Let the lengths of legs be a and b, and the
  length of the hypotenuse be c. A is an acute
  angle.

With reference to angle A, the ratio of the
length of the side opposite angle A to length of
the hypotenuse is defined as sine A
a/c Sine A is abbreviated Sin A. Thus, sin A
a/c.
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The cosine of an acute angle...

• Let the lengths of legs be a and b, and the
  length of the hypotenuse be c. A is an acute
  angle.

With reference to angle A, the ratio of the
length of the side adjacent to angle A to length
of the hypotenuse is defined as cosine A
b/c Cosine A is abbreviated to Cos
A. Thus, cos A b/c.
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The tangent of an acute angle...

• Let the lengths of legs be a and b, and the
  length of the hypotenuse be c. A is an acute
  angle.

With reference to angle A, the ratio of the
length of the side opposite to angle A to length
of the side adjacent to angle A is defined
as tangent A a/b tangent A is
abbreviated Tan A. Thus, tan A a/b
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SOHCAHTOA
S Sine O Opposite H Hypotenuse - C
Cosine A Adjacent H Hypotenuse - T
Tangent O Opposite A Adjacent

• This is a clever technique most people use to
  remember these three basic trig ratios.
• SOH-CAH-TOA sounds strange? What if I told you it
  was the ancient oriental queen who loved
  Geometry? (not true!)

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Example sine, cosine and tangent ratios...

• Find the sine of the given angle. SOHCAHTOA

Sin B Opposite/Hypotenuse sin 53.10 16/20
4/5 0.80 Cos B Adjacent/Hypotenuse Cos 53.10
12/20 3/5 0.60 Tan B
Opposite/Adjacent Tan 53.10 16/12 5/3 1.67
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42
Try it yourself trig ratios...

• Find the value of sine, cosine, and tangent of
  the given acute angle. SOHCAHTOA

Click to choose your answer from the choices
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43
Thats the way!

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• Outstanding !
• SOHCAHTOA would be proud of you.
• You may want to teach others!
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44
You can do it Try again!

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• I am sorry.
• Remember it is SOHCAHTOA all the way!
• You must have used the wrong ratio!
• Click to Review Try Again!

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Angle or Side Lengths?

• Does the trig ratio depend on the size of the
  angle or size of the side length?
• Let us consider similar triangles in our
  investigation.

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46
Find the trig ratios of each...
Remember SOHCAHTOA!

• Compute the ratios and make a conjecture

sin 36.870 ? 0.6 6/10 0.6 9/15 0.6
cos 36.870 ? 0.8 8/10 0.8 12/15 0.8
tan 36.870 ¾ 0.75 6/8 0.75 9/12 0.75
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Conjecture Trigonometric ratios are a property
of similarity (angles) and not of the length of
the sides of a right triangle.
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Computed Trig Ratios ...

• The trig ratios are used so often that technology
  makes these values readily available in the form
  of tables and on scientific calculators.
• We will now show you how to use your calculator
  to find some trig ratios.
• Grab a scientific calculator and try it out.

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How to find trig values on calculator

• Each calculator brand may work a little
  differently, but the results will be the same.
• Look for the trig functions on your calculator
  sin, cos and tan
• select the trig ratio of your choice followed by
  the angle in degrees and execute (enter).
• example sin 30 will display 0.5
• on some calculators you may have to type in the
  angle first then the ratio
• example 30 sin will display 0.5

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Given an angle, use calculator to evaluate trig
ratio

• Use your calculator to verify that the sine,
  cosine and tangent of the following angles are
  correct (to 4 decimals)

Angle A sin A cos A tan A
45o Sin 45o 0.7071 Cos 45o 0.7071 Tan 45o 1.0000
60o Sin 60o 0.8660 Cos 60o 0.5000 Tan 60o 1.7321
30o Sin 30o 0.5000 Cos 30o 0.8660 Tan 30o 0.5774
82.5o Sin 82.5o 0.9914 Cos 82.5o 0.1305 Tan 82.5o 7.5958
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Try it yourself...

• Find the values of the following trig ratios to
  four decimal places

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51
Going backwards finding angles when we know a
trig ratio

• We can use the reverse operation of a trig ratio
  to find the angle with the known trig ratio (n/m)
• The inverse trig ratios are as follows
• Inverse of sin (n/m) is sin-1(n/m)
• Inverse of cos (n/m) is cos-1(n/m)
• Inverse of tan (n/m) is tan-1(n/m)

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Example inverse operation (sin A)

• Suppose we know the trig ratio and we want to
  find the associated angle A.

• From SOHCAHTOA, we know that from the angle A, we
  have the opposite side and the hypotenuse.
• Therefore the SOH part helps us to know that we
  use sin A O/H 4/5
• The inverse is thus sin-1(4/5) A
• A Sin-1 (4/5) 53.13o

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Example inverse operation (cos B)

• Suppose we know the trig ratio and we want to
  find the associated angle B.

• From SOHCAHTOA, we know that from the angle B, we
  have the adjacent side and the hypotenuse.
• Therefore the CAH part helps us to know that we
  use cos A A/H 4/5
• The inverse is thus cos-1(4/5) B
• B cos-1 (4/5) 36.87o

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Example inverse operation (tan A)

• Suppose we know the trig ratio and we want to
  find the associated angle A.

• From SOHCAHTOA, we know that from the angle A, we
  have the opposite side and the adjacent side.
• Therefore the TOA part helps us to know that we
  use tan A O/A 4/3
• The inverse is thus tan-1(4/3) A
• A tan-1 (4/3) 53.13o

3
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Try it yourself(use inverse)

• Use a calculator to find the measure of the
  angles A and B.

Use SOHCAHTOA as a guide to what ratio to use.
m?A ? A. 38.7 B. 51.3 C. 53.1
m?B ? A. 38.7 B. 51.3 C. 53.1
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Perfect!

• ?
• You make me smile!
• sin-1(x) is also referred to as arcsin(x)
• cos-1(x) is also referred to as arccos(x)
• tan-1(x) is also referred to as arctan(x)

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Looks like youre in trouble!

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• I am sorry.
• Not to worry, I can help.
• Click to GO BACK, REVIEW TRY Again!

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Finding the legs of a right

• Use trig ratios to find sides of a triangle.
  Remember SOHCAHTOA!

• With reference to angle A,
• b is the length of side adjacent and
• a is the length of the side opposite the angle.
• the hypotenuse is given
• Strategy make an equation that uses only one leg
  and the hypotenuse at a time.

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59
Finding the legs of a right

• The tangent ratio may not easily help you figure
  out the legs a and b in this case. (SOHCAHTOA!)

Using tangent tan A O/A Substituting values
from the tgriangle tan 30o a/b From the
calculator tan 30o 0.5774 Thus tan 30o a/b
0.5774 a/b And, a 0.5774(b) GETS YOU
STUCK!
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60
Finding the legs of a right

• Using sine ratio to find the leg of a triangle.
  Remember SOHCAHTOA!

Using sine sin A O/H Substituting values
from the tgriangle Sin 30o a/12 From the
calculator sin 30o 0.5 Thus sin 30o a/12
0.5 a/12 And a 0.5(12) 6
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Finding the legs of a right

• Using the cosine ratio to find legs of a
  triangle. Remember SOHCAHTOA!

Using cosine cos A A/H Substituting values
from the tgriangle cos 30o b/12 From the
calculator cos 30o 0.8660 Thus cos 30o b/12
0.866 b/12 And b 0.866(12) 10.39
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Try it yourself legs

• Find the lengths of the legs of the triangle and
  the third angle. Choose the correct answer.

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63
Having trouble?

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• I am sorry you are having problems.
• Not to worry, I can help.
• Click to GO BACK, REVIEW TRY Again!

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You got it!

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• Fantastic !
• You are on the right track.
• Click Go BACK and CONTINUE

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Finding the hypotenuse ...

• Use trig ratios to find the hypotenuse of a
  triangle. Remember SOHCAHTOA!

• With reference to angle A,
• b is the length of side adjacent and
• 12 is the length of the side opposite the angle.
• c is the hypotenuse
• Strategy make an equation that uses only one
  unkown at a time.

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66
Finding the hypotenuse ...

• Use trig ratios to find the hypotenuse of a
  triangle. Remember SOHCAHTOA!

Since 12 is opposite to the angle, we use the
sine ratio Sine A O/H Substituting values
from the tgriangle sin 30o 12/c From the
calculator sin 30o 0.5 Thus sin 30o 12/c
or 0.5 12/c c 12/0.5 24
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Try it yourself hypotenuse

• Find the lengths of the hypotenuse, leg b and the
  third angle. Choose the best answer.

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Terrific!

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• I am proud of your progress!
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Help is a click away!

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• Review and try again.
• PLEASE GO BACK, REVIEW TRY again.

70
Putting it all together

• We now have the tools we need to solve any right
  triangle (to determine the lengths of each and
  all sides and the angles, given minimal
  information) Remember SOHCAHTOA!
• Typically you get two pieces of information
• One side length and one angle or
• Two sides lengths

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Solve the triangle (side angle)

• Given one side length and one angle, determine
  the rest. Remember SOHCAHTOA !

Find measure of angle B and side lengths AC and
AB.
Since we know two angles (90 and 42) we can
determine the 3rd from the Triangle Angle Sum
Theorem m?B 1800 (900420) 480.
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72
Solve the triangle (side angle)

• Given one side length and one angle, determine
  the rest. Remember SOHCAHTOA !

Strategy side with length 12 is opposite to
angle A. To find b, use tan A and to find c, use
sin A
sin A O/H sin 42 12/c 0.6691 12/c c
12/0.6691 c 17.93
tan A O/A tan 42 12/b 0.9004 12/b b
12/0.9004 b 13.33
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Try it yourself side angle

• Solve the triangle. Choose and check answer.

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Congratulations!

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• Its fun when you get it!
• You are on the right track.
• Click Go BACK and CONTINUE

75
Sorry, not correct!

• ?
• I am sorry.
• Remember it is SOHCAHTOA all the way!
• You must have used the wrong ratio!
• Click to Review Try Again!

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Solve the triangle (2 sides)

• Given two side lengths, solve the triangle.
  Remember SOHCAHTOA !

• Strategy
• use Pythagorean Theorem to find the 3rd side
  length, a.
• Use cosine ratio to find measure of angle A
• Use the Triangle Angle Sum Theorem to find the
  measure of angle B.

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Solve the triangle (2 sides)

• Given two side lengths, solve the triangle.
  Remember SOHCAHTOA !

• Using Pythagorean Theorem to find the 3rd side
  length, a.
• c2 a2 b2 Pythagorean Theorem
• 172 a2 102 Substituting values
• 289 a2 100 Evaluating the squares
• a2 289-100 Addition property of
• a2 189 Simplifying
• a sqrt(189) 13.75 Taking square root.

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Solve the triangle (2 sides)

• Given two side lengths, solve the triangle.
  Remember SOHCAHTOA !

• Using cosine ratio to find measure of angle A
• cos A A/H (the CAH part)
• cos A 10/17 (substituting values)
• m?A cos-1(10/17) (inverse of cosine)
• m?A 53.97o (Calculator)

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Try it yourself(2 sides)

• Solve the triangle. Click to check your answer

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You have mastered lot!

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• Congratulations !
• We are almost done!
• You did it again!
• Click Go BACK and CONTINUE

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Sorry you got it wrong!

• ?
• I am sorry.
• Go back and try again.
• A little review will surely help.
• Click to Review Try Again!

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Real life Applications

• Trigonometry is used to solve real life problems.
• The following slides show a few examples where
  trigonometry is used.
• Search the Internet for more examples if you
  like.

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Real life example 1

• Measuring the height of trees

What would you need to know in order to calculate
the height of this tree? What trig ratio would
you use?
Click here to see if we agree.
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84
Finding height of towers..

• Tall buildings (skyscrapers), towers and
  mountains

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85
Solve real problem

• Assume the line in the middle of the drawn
  triangle is perpendicular to the beach line.
• How far is the island from the beach?

Click here to check my solution and compare with
yours
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86
Distance to the beach(Ans.)

• The distance we want is the shortest distance
  ?.
• The tangent ratio can be used here
• tan 30o x/50
• 0.57735 x/50
• x 0.57735(50) 28.8675
• Therefore the island is about 29m from the beach.

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Real life example 1 Answer

• h is the height of the tree. That is what we are
  looking for.
• We need to know the angle of elevation ? and
  also the horizontal distance from A to the bale
  od the tree, x
• The tangent ratio would be used
• tan ? h/x and so h xtan ?

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Congratulations

• Be proud of yourself. You have successfully
  completed a crash course in basic trigonometry
  and I expect you to be able to do well on this
  strand in the Common Core States Standards test.
  Print the certificate to show your achievement.

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89
Certificate of Completion

• I hereby certify that
• _____________________________________
• has satisfactorily completed a basic course in
  Introduction to Trigonometry on this day the
  ____________________ of the year 20___
• The bearer is qualified to solve some real world
  problems using trigonometry.
• Signed Nevermind E. Chigoba

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