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I. Basic Facts and Definitions 1. Right angle – angle measuring 90

2. Straight angle – angle measuring 180

3. Acute angle – angle measuring between 0 and 90

4. Complementary angles – two angles whose sum is 90

5. Supplementary angles – two angles whose sum is 180

6. Right triangle – triangle with a right angle

7. Isosceles triangle – a triangle with exactly two sides equal

8. Equilateral triangle – a triangle with all three sides equal

9. The sum of the angles of a triangle is 180 .

10. In general, capital letters refer to angles while small letters refer

to the sides of a triangle. For example, side a is opposite angle A

.

II. Right Triangle Facts and Examples

1. Hypotenuse – the side opposite the right angle, side c .

2. Pythagorean Theorem - 222 cba

3. A and B are complementary.

Right Angle Trigonometry

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III. Examples: 1. In a right triangle, the hypotenuse is 10 inches and one side is 8

inches. What is the length of the other side?

Solution: 222 cba

222 108 b

10064 2 b

362 b

6b

2. In a right triangle ABC , if 23A , what is the measure of B ?

Solution: The two acute angles in a right triangle are

complementary. 90BA

9023 B

67B

IV. Similar Triangles: a. Two triangles are similar if the angles of one triangle are equal to

the corresponding angles of the other. In similar triangles, ratios of corresponding sides are equal.

B

A C

8a

?b

10c

C

B

A

Conditions for Similar Triangles (EGFABC ~ )

1. Corresponding angles in similar triangles are equal:

EA FB GC

2. Ratios of corresponding sides are equal:

FG

BC

EF

AB

EG

AC

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Example 1:

Example 2:

B C

A

F E

22

50

50

50AE meters 22EF m and 100AB m

Find the length of side BC .

Notice that ABC and AEF are similar since corresponding angles are equal. (There

is a right angle at both F and C , A is the

same in both triangles and B equals the acute angle at E .)

Solution: BC

EF

AB

AE so

BC

22

100

50

By cross multiplying we get:

)100(22)(50 BC

Therefore 44BC meters.

50AE meters 22EF m and 100AB m

Find the length of side BC .

All 904545 triangles are similar to one another. Two sides are of equal length and the

hypotenuse is 2 times the length of each of the equal sides.

All 906030 triangles are similar to one another. The shortest side of length a is opposite

the smallest angle ( 30 ). The hypotenuse is twice the length of the shortest side. The side opposite

the 60 has a length 3 times the shorter leg.

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Problem: Find the lengths of the legs of a 906030 triangle if the hypotenuse is 8 meters.

Solution: 1) If 82 a , then 4a meters and 2) 34)4(33 a meters.

V. The Six Trigonometric Ratios for Acute Angles TRIG TRICK: A good way to remember the trig ratios is to use the mnemonic SOH CAH TOA!

sine Ac

a

hypotenuse

oppositeA sin cosecant A

a

c

opposite

hypotenuse

AA

sin

1csc

cosine Ac

b

hypotenuse

adjacentA cos secant A

b

c

adjacent

hypotenuse

AA

cos

1sec

tangent Ab

a

adjacent

oppositeA tan cotangent A

a

b

opposite

adjacent

AA

tan

1cot

c

a

b A

B

C

SOH CAH TOA ine

pposite

ypotenuse

osine

djacent

ypotenuse

angent

pposite

djacent

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Example 1:

Find the six trigonometric ratios for the acute angle B . Solution:

c

b

hypotenuse

oppositeB sin . Using the above definitions, the rest are:

c

aB cos ,

a

bB tan ,

b

cB csc ,

a

cB sec ,

b

aB cot

Example 2:

In the right ABC , 1a and 3b . Determine the six trigonometric ratios for

B . Solution: Use Pythagorean Theorem:

222 bac 222 31 c

22 10c

10c

(Since length is positive, we will only use 10c .)

10

103

10

3sin

hyp

oppB

10

10

10

1cos

hyp

adjB 3

1

3tan

adj

oppB

3

10csc

opp

hypB 10

1

10sec

adj

hypB

3

1cot

opp

adjB

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VI. Special Cases

a. Trigonometric values of 30 and 60 (Use the 906030 triangle from pg. 3.)

b. Trigonometric values of 45 (Use the 904545 triangle from pg. 3.)

1b

2c 3a

60

30 2

130sin

2

360sin

2

330cos

2

160cos

3

3

3

130tan 3

1

360tan

21

230csc

3

32

3

260csc

3

32

3

230sec 2

1

260sec

31

330cot

3

3

3

160cot

1b

1a 2c

45

45

30

60

2

2

2

145sin 2

1

245csc

2

2

2

145cos 2

1

245sec

11

145tan 1

1

145cot

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VII. Converting Minutes and Seconds to Decimal Form

(Necessary for most calculator use in evaluating trig values) 1. To convert from seconds to a decimal part of a minute, divide the

number of seconds by 60. 2. To convert from minutes to a decimal part of a degree, divide the

number of minutes by 60.

Example 1: Convert 7464 to degrees using decimals.

Solution:

60

47647464

783.64783.647464

Example 2: Convert 012115 to degrees using decimals.

Solution: 012115012115

60

102115012115

716.1215012115

60

167.1215012115

203.15203.15012115

VIII. Right Triangle Trigonometry Problems

To Solve Right Triangle Problems: There are six parts to any triangle; 3 sides and 3 angles. Each trig formula (ex: sin A = a/c) contains three parts; one acute angle and two sides. If you know values for two of the three parts then you can solve for the third unknown part using the following method:

1. Draw a right triangle. Label the known parts with the given values and indicate the unknown part(s) with letters.

2. To find an unknown part, choose a trig formula which involves the unknown part and the two known parts.

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Example 1: A right triangle has 38a and 61B . Find the length of side b .

Solution: (Always check your answer by comparing size of angle and length of side; the longer side is always opposite the larger angle.)

IX. Angles of Elevation and Depression

Example: From a point 124 feet from the foot of a tower and on the same level, the angle of

elevation of the tower is 0236 . Find the height of the tower. Solution:

Angle of elevation

61

b

38a B

A

C

Which trig formulas involve an acute angle (B) and the side opposite (b) and the side adjacent (a) to the angle? Since both tangent and cotangent do, either could be used to solve this problem. We will use tangent.

a

b

adj

oppB tan so,

3861tan

b or

388040.1

b .

Therefore 6.68b .

Angle of depression

h

124 ft.

0236

)333.36tan(0236tan24

h

)7355.0(124h

2.91h ft.

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Practice Problems:

1. In right triangle ABC , if 39c inches and 36b inches, find a .

2. Find the length of side AC . Note: This problem and diagram corresponds to finding the height of a street light pole ( AC ) if a 6 ft. man ( EF ) casts a shadow ( BF ) of 15 ft. and the pole casts a shadow ( BC ) of 45 ft.

3. Evaluate:

a) sin E = _____________ b) tan E = _____________

c) cos F = _____________ d) sec F = _____________

4. Evaluate: (Draw reference 906030 and 904545 triangles)

a) sin 30 = _____________ b) tan 60 = _____________

c) sec 60 = _____________ d) tan 45 = _____________

e) csc 45 = _____________ f) cot 30 = _____________

B C

A

E

F

6

15 30

F

G E

13 12

5

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5. Evaluate:

a) tan A = _____________ b) csc B = _____________

c) cot A = _____________ d) sec B = _____________

6. Convert to decimal notation using a calculator:

a) 6076

b) 317245 Evaluate, using a calculator:

c) 8452sin

d) 2439cot

7. Label the sides and remaining angles of right triangle ABC , using A , B ,

a , b and c . If 43a and 37A , find the values of the remaining parts.

B

C A

10 8

C

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8. Given right triangle ABC with 622.0b and 0451 A , find c . Draw a diagram.

9. From a cliff 140 feet above the shore line, an observer notes that the angle

of depression of a ship is 0321 . Find the distance from the ship to a point on the shore directly below the observer.

cliff ship

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Answers to Right Triangle Trigonometry: 1. 15a inches (use Pythagorean Theorem)

2. BF

BC

EF

AC

15

6

45

AC 18AC

3. a) 13

12sin E b)

5

12tan E c)

13

12cos F d)

12

13sec F

4. (see part E of the handout)=

a) 2

130sin b) 360tan c) 260sec

d) 145tan e) 245csc f) 330cot

5. 6b (use Pythagorean Theorem)

a) 3

4

6

8tan A b)

3

5csc B c)

4

3cot A d)

4

5sec B

6. a) 1.76 b) 454.45 c) 7965. d) 2045.1 7. 53B 06.57b 45.71c

8. 1c (Use c

bA cos or

b

cA sec to solve for unknown c )

9. x

1400321tan

41.355x ft.

C

B

A b

a c

cliff ship

(Angle of depression)