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1.1 ANGLES

An angle consists of two rays in a plane with a common endpoint. The two rays are called the sides of the angle, and the common endpoint is called the vertex.

The degree is a unit of measurement for angles. There are 360° degrees in a circle. A half circle has 180° and a right angle has 90°.

1.2 DEGREE OF MEASURE

If an angle has measure greater than 0° and less than 90°, then it is called an acute angle.

If an angle has measure greater than 90°and less than 180°, then it is called an obtuse angle.

If an angle has measure equal to 90°, then it is called a right angle.

An angle that has measure equal to 180° is called a straight angle.

If the sum of the measures of two positive angles is 90°, then the angles are called complementary angles.

If the sum of the measures of two positive angles is 180°, then the angles are called supplementary angles.

1.2 DEGREE OF MEASURE

Finding an Angle in a Right Angled Triangle

We can find an unknown angle in a right-angled triangle, as long as we know the lengths of two of its sides. Example, 5ft ladder leans against a wall as shown. What is the angle between the ladder and the wall?

The answer is to use "SOHCAHTOA"and find the names of the two sides you knowAdjacent is adjacent to the angle,Opposite is opposite the angle, and the longest side is the Hypotenuse.

6

45°-45°-90° Special Right TriangleIn a triangle 45°-45°-90° , the hypotenuse is times as long as a

leg.

2

2

45°

45°

Hypotenuse

XX

X

Leg

Leg

Example:

45°

45°

5 cm

5 cm

5 cm

2

Right Triangle

7

30°-60°-90° Special Right TriangleIn a triangle 30°-60°-90° , the hypotenuse is twice as long

as the shorter leg, and the longer leg is times as long as the shorter leg.

30°

60°

Hypotenuse

3X

2X

X

Longer Leg

Shorter Leg

Example:

30°

60°

10 cm

5 cm

3

5 cm3

Right Triangle

Example: Find the value of a and b.

60°7 cm

a

b

Step 1: Find the missing angle measure. 30°

30 °

Step 2: Decide which special right triangle applies. 30°-60°-90°Step 3: Match the 30°-60°-90° pattern with the problem.

30°

60°x

2x3x

a = cm

b = 14 cm

Step 5: Solve for a and b

7 3

Step 4: From the pattern, we know that x = 7 , b = 2x, and a = x .3

Example: Find the value of a and b.

2

45°7 cm

a

b

Step 1: Find the missing angle measure. 45°

45 °

Step 2: Decide which special right triangle applies. 45°-45°-90°

Step 3: Match the 45°-45°-90° pattern with the problem.

45°

45°x

xx

Step 4: From the pattern, we know that x = 7 , a = x, and b = x .

a = 7 cm

b = 7 cm

Step 5: Solve for a and b

2

2

2

When a person looks at something above his/her location, the angle between the line of sight and the horizontal is called the angle of elevation. In this case, the line of sight is "elevated" above the horizontal.

Eye

Object

Angle of Elevation & Angle of Depression

When a person looks at something below his or her location, the angle between the line of sight and the horizontal is called the angle of depression. In this case, the line of sight is "depressed" below the horizontal.

Eye

Object


A stands at the window of a so that his are 12.6 m above thelevel ground in the vicinity of the

An object is 58.5 m away from the on a line directly beneath the

Find the angle of depression of the person's line of sight to the object on the ground.


Solutionsq = angle of depression

12.6tan 58.5q

eye12

.6m

Ground levelobject

58.5m

Line of sight

q

1 12.6tan 58.5q

q 12.15º

q

FIND THE HEIGHT OF THE TOWER

80º

53m

Exercise 2

Height of tower = 53 tan 80°= 300.6m

Exercise 1

Calculate the angle of elevation of the line of sight of a person whose eye is 1.7 m above the ground, and is looking at the top of a tree which is 27.5 m away on level ground and 18.6 m high.

16.9tan 27.5q

1 16.9tan 27.5q

q 31.57ºAngle of elevation of this person line of sight is 31.57º

Angle of depression

100m

h

Find the height of the man in the hot air balloon from the ground.

= 50°h =100 tan 50° = 119.2m

2. RIGHT TRIANGLE

3.1 Oblique Triangles and the Law of Sines

A triangle that is not a right triangle is called an oblique triangle. We can solve oblique triangles using the Law of Sines and the Law of Cosines.

Law of Sine 1. Two angles and any side (AAS or ASA) 2. Two sides and an angle opposite one of them (SSA)Law of Cosine 3. Three sides (SSS) 4. Two sides and their included angle (SAS)

Law of Sines and Cosines

Example 1

Triangle in figure, C = 102°, B = 29°, and b = 28 feet. Find the remaining angle and sides.

Given Two Angles and One Side—AAS

The third angle of the triangle is

A = 180° – B – C = 180° – 29° – 102°= 49°

Law of SineWhat law should be applied?

Solution:Angle properties of triangles

We already know that the angles in a triangle add up to 180°. Angles on a straight line also add up to 180°.

Example 1

Find the value of aFind the value of c

Example 2

For the triangle in Figure, a = 22 inches, b = 12 inches, and A = 42°. Find the remaining side and angles.

Given Two Sides and One Angle—SSA

Law of SineWhat law should be applied?Find the value of B

Example 2

For the triangle in Figure, a = 22 inches, b = 12 inches, and A = 42°. Find the remaining side and angles.

Given Two Sides and One Angle—SSA

Now, you can determine that C = 180° – 42 ° – 21.41 ° = 116.59 °

Then, Find the value of c

Sometimes the sine law is not enough to help us solve for a non-right angled triangle.For example:

C

BA

a14

18300

In the triangle shown, we do not have enough informationto use the sine law. It only provided the following

CBa

sin18

sin14

30sin 0

Where there are too many unknowns.

For this reason we derive another useful result, known as the COSINE RULE. The Cosine Rule maybe used when:a. Two sides and an included angle are given.b. Three sides are given

B

C

A

a

b

c

C

B

A

ac

To find the length of a sidea2 = b2+ c2 - 2bc cos Ab2 = a2 + c2 - 2ac cos Bc2 = a2 + b2 - 2ab cos C

To find an angle when given all three sides.

bcacb

A2

cos222

ac

bcaB

2cos

222

abcba

C2

cos222

Example 3

Find the three angles of the triangle in figureThree Sides of a Triangle—SSS

Solution:First find the angle opposite the longest side—side b in this case. Using the alternative form of the Law of Cosines, you find that

Law of CosineWhat law should be applied?

Example 3

You know that A must be acute because B is obtuse, and a triangle can have, at most, one obtuse angle.

So, A = 22.08°

and C = 180 ° – 22.08 ° – 116.80 ° = 41.12 °

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