sohcahtoa passport to trigonometry land... click to continue

SOHCAHTOApassport to Trigonometry Land ...

CLICK TO CONTINUE

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.

CLICK TO CONTINUE

Examples of triangles...

All these polygons are tri-gons and commonly called triangles

CLICK TO CONTINUE

These are not triangles...None of these is a

triangle...

Can you tell why not?CLICK TO CONTINUE

Try it yourself...

Click on each one that IS a triangle?

CLICK TO CONTINUE

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.

CLICK TO CONTINUE

These are right triangles...

CLICK TO CONTINUE

All right triangles

These are not right triangles...

These triangles are NOT right triangles. Explain why not?

CLICK TO CONTINUE

Try it yourself ...

Click on the triangle that is NOT a right triangle?

CLICK TO CONTINUE

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.

CLICK TO CONTINUE

Parts of a right triangle ...

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

hypotenuseleg

legCLICK TO CONTINUE

Try it yourself … hypotenuse

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

side 2

side 1

side 3

CLICK TO CONTINUE

Try it yourself … shorter leg

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

side 2

side 1

side 3

CLICK TO CONTINUE

Try it yourself … longer leg

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

side 2

side 1

side 3

CLICK TO CONTINUE

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.

hypotenuseleg1

leg2

c

a

b

If the lengths of hypotenuse and legs are c,

a and b respectively, then c2 = a2 + b2 CLICK TO CONTINUE

Finding a side - hypotenuseUse the

Pythagorean Theorem to find the length of the missing side.

14

10x c2 = a2 + b2

x2 = 102 + 142

= 100 + 196= 296

x = sqrt(296)= 17.2

CLICK TO CONTINUE

Finding a side … legUse the

Pythagorean Theorem to find the length of the missing side.

x

1015

c2 = a2 + b2 152 = 102 + x2

225 = 100 + x2

x2 = 125x = sqrt(125)

= 11.18

CLICK TO CONTINUE

Try it yourself- hypotenuseFind the hypotenuse

of the given right triangle with the lengths of the legs known:

x

8

6

Click on the selection that matches your answer:

A. 36

C. 100

B. 10

D. 64

CLICK TO CONTINUE

Try it yourself… legFind the leg of the

given right triangle with the lengths of the leg and hypotenuse known:

15

9

x

Click on the selection that matches your answer:

A. 24

C. 144

B. 6

D. 12

CLICK TO CONTINUE

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.

hypotenuseleg1

leg2

c

a

b

A

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

CLICK TO CONTINUE

Opposite side (to an acute angle)


hypotenuseleg1

leg2

c

a

b

A

leg2 is opposite to acute angle Aleg1 is NOT opposite to acute angle A

CLICK TO CONTINUE

Adjacent side (to an acute angle)


leg1 is adjacent to acute angle Aleg2 is NOT adjacent to acute angle A

hypotenuseleg1

leg2

c

a

b

A

CLICK TO CONTINUE

Try it yourself … opposite

Click on the side that is opposite to angle B.

hypotenuseleg1

leg2

c

a

b

B

CLICK TO CONTINUE

Try it yourself … adjacent

Click on the side that is adjacent to angle B. hypotenuseleg1

leg2

c

a

b

BCLICK TO CONTINUE

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

CLICK TO CONTINUE

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.

hypotenuseleg1

leg2

c

a

b

AWith 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/cSine A is abbreviated Sin A.Thus, sin A = a/c.

length of side opposite A length of the hypotenuse

CLICK TO CONTINUE

The cosine of an acute angle...


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.

length of side adjacent to angle A length of the hypotenuse

hypotenuseleg1

leg2

c

a

b

A

CLICK TO CONTINUE

The tangent of an acute angle...


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

length of side opposite to angle A length of the adjacent side

hypotenuseleg1

leg2

c

a

b

A

CLICK TO CONTINUE

SOHCAHTOA

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!)

S = SineO = OppositeH = Hypotenuse-C = CosineA = AdjacentH = Hypotenuse-T = TangentO = OppositeA = Adjacent CLICK TO CONTINUE

Example… sine, cosine and tangent ratios...

Find the sine of the given angle. [SOHCAHTOA]

53.10

2016

12B

Sin B = Opposite/Hypotenusesin 53.10 = 16/20 = 4/5 = 0.80

Cos B = Adjacent/HypotenuseCos 53.10 = 12/20 = 3/5 = 0.60

Tan B = Opposite/AdjacentTan 53.10 = 16/12 = 5/3 =1.67

CLICK TO CONTINUE

Try it yourself … trig ratios...Find the value of sine, cosine, and tangent of

the given acute angle. [SOHCAHTOA]

53.10

15

12

9

B Click to choose your answer from the choices

cos 53.10 =? A. 5/3 B. 3/5 C. 4/3 D. 3/4 E. 4/5 F. 5/4

sin 53.10 =? A. 5/3 B. 3/5 C. 4/3 D. 3/4 E. 4/5 F. 5/4

tan 53.10 =? A. 5/3 B. 3/5 C. 4/3 D. 3/4 E. 4/5 F. 5/4

CLICK TO CONTINUE

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.

54

3

36.870

A

108

6

36.870

A

1512

9

36.870

A

CLICK TO CONTINUE

Find the trig ratios of each...

Compute the ratios and make a conjecture

54

3

36.870

A

108

6

36.870

A

1512

9

36.870

A

sin 36.870 ⅗ = 0.6 6/10 = 0.6 9/15 = 0.6

cos 36.870 ⅘ = 0.8 8/10 = 0.8 12/15 = 0.8

tan 36.870 ¾ =0.75 6/8 = 0.75 9/12 = 0.75Conjecture: Trigonometric ratios are a property of similarity (angles)

and not of the length of the sides of a right triangle.

[Remember SOHCAHTOA!]

CLICK TO CONTINUE

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.

CLICK TO CONTINUE

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).o example: sin 30 will display 0.5

• on some calculators you may have to type in the angle first then the ratioo example: 30 sin will display 0.5

CLICK TO CONTINUE

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

Asin 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 =7.5958CLICK TO CONTINUE

Try it yourself...

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

CLICK TO CONTINUE

sin 34o = ? A. 0.8290 B. 0.6745 C. 0.5592

cos 56o = ? A. 0.5592 B. 0.8290 C. 1.4826

tan 72o = ? A. 0.3090 B. 0.9511 C. 3.0777

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) CLICK TO CONTINUE

Example inverse operation (sin A) Suppose we know the trig ratio and we want

to find the associated angle A.

54

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

CLICK TO CONTINUE

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

54

B

A

CLICK TO CONTINUE

Example inverse operation (tan A) Suppose we know the trig ratio and we want

to find the associated angle A.

4

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) = 53.13o3

CLICK TO CONTINUE

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.

CLICK TO CONTINUE

A

B

C

12

15

19.21 mA =? A. 38.7 B. 51.3 C. 53.1

mB =? A. 38.7 B. 51.3 C. 53.1

Finding the legs of a right Use trig ratios to find sides of a triangle.

Remember SOHCAHTOA!

300

A

12a

b

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.

CLICK TO CONTINUE

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!)

300

A

12a

b

Using tangent: tan A = O/A Substituting values from the tgriangle:tan 30o = a/bFrom the calculator: tan 30o = 0.5774Thus tan 30o = a/b

0.5774 = a/bAnd, a = 0.5774(b) GETS YOU STUCK!

CLICK TO CONTINUE

Finding the legs of a right

Using sine ratio to find the leg of a triangle.

Remember SOHCAHTOA!

300

A

12a

b

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

CLICK TO CONTINUE

Finding the legs of a right Using the cosine ratio to find legs of a

triangle. Remember SOHCAHTOA!

300

A

12a

b

Using cosine: cos A = A/H Substituting values from the tgriangle:cos 30o = b/12From the calculator: cos 30o = 0.8660Thus cos 30o = b/12

0.866 = b/12And b = 0.866(12) = 10.39

CLICK TO CONTINUE

Try it yourself… legsFind the lengths of the legs of the triangle and the third angle. Choose the correct answer.

21

25oC

B

A

a

b

CLICK TO CONTINUE

mB = ? A. 90 B. 65 C. 25

a = ? A. 10.57 B. 22.66 C. 21

b = ? A. 21 B. 10.57 C. 22.66

Finding the hypotenuse ... Use trig ratios to find the hypotenuse of a


300

A

c12

b

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.

CLICK TO CONTINUE

Finding the hypotenuse ... Use trig ratios to find the hypotenuse of a


300

A

c12

b

Since 12 is opposite to the angle, we use the sine ratio: Sine A = O/HSubstituting values from the tgriangle:

sin 30o = 12/cFrom the calculator: sin 30o = 0.5Thus sin 30o = 12/c or 0.5 = 12/c

c = 12/0.5 = 24

CLICK TO CONTINUE

Try it yourself… hypotenuse

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

c

C

B

A

13

b

55o

CLICK TO CONTINUE

mA = ? A. 35 B. 45 C. 55

b = ? A. 22.66 B. 10.57 C. 18.57

c = ? A. 18.57 B. 10.57 C. 22.66

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

CLICK TO CONTINUE

Solve the triangle (side + angle)

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

B

420

A

12

C

c

b

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: mB = 1800 –(900+420) = 480.

CLICK TO CONTINUE

Solve the triangle (side + angle)

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

420

A

12

B

C

c

b

Strategy: side with length 12 is opposite to angle A. To find b, use tan A and to find c, use sin A

tan A = O/Atan 42 = 12/b0.9004 = 12/bb = 12/0.9004b = 13.33

sin A = O/Hsin 42 = 12/c0.6691 = 12/cc = 12/0.6691c = 17.93

CLICK TO CONTINUE

Try it yourself… side + angle Solve the triangle. Choose and check

answer.

440

A

23

B

C

c

b

CLICK TO CONTINUE

mB = ? A. 46 B. 44 C. 23

c = ? A. 33.11 B. 23.82 C. 23

b = ? A. 23 B. 23.82 C. 33.11

Solve the triangle (2 sides)Given two side lengths, solve the triangle.

Remember SOHCAHTOA !

C

A

B

17

a

10

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.

CLICK TO CONTINUE

Solve the triangle (2 sides) Given two side lengths, solve the triangle.


BC

A

17

a

10

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. CLICK TO CONTINUE

Solve the triangle (2 sides)Given two side lengths, solve the triangle.


CA

B

17a

10

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

CLICK TO CONTINUE

Try it yourself…(2 sides)Solve the triangle. Click to check your

answer…

B

45

18

C A

c

CLICK TO CONTINUE

mA = ? A. 21.8 B. 68.2 C. 48.5

c = ? A. 21.8 B. 68.2 C. 48.5

mB = ? A. 21.8 B. 68.2 C. 48.5

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.

CLICK TO CONTINUE

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.CLICK TO CONTINUE

Finding height of towers.. Tall buildings (skyscrapers), towers and

mountains…

CLICK TO CONTINUE

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 yoursCLICK TO CONTINUE

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.

CLICK TO CONTINUE

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

sohcahtoa passport to trigonometry land... click to continue

Documents