passport to the world of Trigonometry CLICK TO CONTINUE

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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All these polygons are tri-gons and commonly called triangles

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None of these is a triangle...

Can you tell why not?CLICK TO CONTINUE

CLICK on each one that IS a triangle?

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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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Can you tell why?

These triangles are NOT right triangles. Explain why not?

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Click on the triangle that is NOT a right triangle?

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

hypotenuseleg

legCLICK TO CONTINUE

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

side 2

side 1

side 3

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

side 2

side 1

side 3

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

side 2

side 1

side 3

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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

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Use 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

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Use 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

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Find 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

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Find 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

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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

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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

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

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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 Aleg2 is NOT adjacent to acute angle A

hypotenuseleg1

leg2

c

a

b

A

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Click on the side that is opposite to angle B.

hypotenuseleg1

leg2

c

a

b

B

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Click on the side that is adjacent to angle B.

hypotenuseleg1

leg2

c

a

b

BCLICK TO CONTINUE

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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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

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

length of side opposite A length of the hypotenuse

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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.

length of side adjacent to angle A length of the hypotenuse

hypotenuseleg1

leg2

c

a

b

A

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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

length of side opposite A length of the adjacent

hypotenuseleg1

leg2

c

a

b

A

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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

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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

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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

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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

A10

8

6

36.870

A

1512

9

36.870

A

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• 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 = 0.6cos 36.870 ⅘ = 0.8 8/10 = 0.8 12/15 =

0.8tan 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!]

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• 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

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 0.5CLICK TO CONTINUE

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

=7.5958CLICK TO CONTINUE

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

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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

• We can use the inverse 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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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

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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

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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

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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

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

300A

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 givenStrategy: make an equation that uses only one leg and the hypotenuse at a time.

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

300A

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!

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

300A

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

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

300A

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

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Find 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

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

300A

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 hypotenuseStrategy: make an equation that uses only one unkown at a time.

CLICK TO CONTINUE

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

300A

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

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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

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

B

420A

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

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

420A

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

Solve the triangle. Choose and check answer.

440A

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

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.

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

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.

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

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)

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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

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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Measuring the height of treesWhat 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

Tall buildings (skyscrapers), towers and mountains…

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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

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

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