MAT 129 Precalculus Trigonometry Review Angles and their Measures - - PowerPoint PPT Presentation

MAT 129 ‐ Trig 1

MAT 129 – Precalculus Trigonometry Review

David J. Gisch

Angles and their Measures

Angles Angles

MAT 129 ‐ Trig 2

Angles and Quadrants Degrees and Radians

• A radian is the size of an angle that subtends an arc, the

length of the radius.

• It turns out that the size of the radius does not matter.

Radians

Recall that the circumference of a circle is 2. If the radius is 1, the circumference is 2 (i.e. the arc length of the whole circle is 2).

360° 2 180° 2

• Radians/ Degrees

MAT 129 ‐ Trig 3

Radians/ Degrees Arc Length

For a circle of radius , a central angle of radians subtends and arc whose length is

Arc Length

Example 7.1.1: Find the length of the arc of a circle of radius 4 meters subtended by a central angle of 0.5 radian.

Arc Length

Example 7.1.2: Find the length of the arc of a circle of radius 10 meters subtended by a central angle of 30°.

MAT 129 ‐ Trig 4

Area of a S ector

The area of the sector of a circle with radius formed by a central angle of radians is 1 2

Arc Length

Example 7.1.3: Find the area of the sector of a circle of radius 5 feet formed by an angle of 40°.

Angular/ Linear S peed

Linear Speed Angular Speed

• S

peed

Example 7.1.4: A child is spinning a rock at the end of a 2-foot rope at the rate of 180 revolutions per minute (rpm). Find the linear speed of the rock when it is released.

MAT 129 ‐ Trig 5

S peed

Example 7.1.5: The diameter of each wheel of a bicycle is 26

• inches. If you are traveling at a speed of 35 miles per hour on

this bicycle, how many revolutions per minute are the wheels turning.

Right Triangle Trigonometry

Pythagorean Theorem Right Triangle Trig

MAT 129 ‐ Trig 6

Right Triangle Trig Right Triangle Trig

Example 7.2.1: Find the value of each of the six trigonometric functions of the angle .

Right Triangle Trig Inverse Functions

MAT 129 ‐ Trig 7

Right Triangle Trig

Example 7.2.2: A person whose eyes are 5 feet above the ground is standing on the runway of an airport 100 feet from the control tower. That person observes an air traffic controller at the window of the 132-foot tower. What is the angle of elevation?

Special Triangles

45-45-90

• If you have a right triangle with angles of 45° the sides

are always in the ratio of ℓ ℓ ℓ 2

30-60-90

• If you have a right triangle with angles of 30° and 60° the

sides are always in the ratio of ℓ ℓ 2 2ℓ

MAT 129 ‐ Trig 8

S pecial Triangles Right Triangle Trig

Example 7.3.1:

2 2

Find the exact value of each expression. (a) sin tan (b) sin60 cos45 (c) cos cot 3 4 3 4        

The Unit Circle

Recall S pecial Triangles

MAT 129 ‐ Trig 9

Unit Circle Unit Circle

1 0, 1 1, 0 30°

Unit Circle THE Unit Circle

MAT 129 ‐ Trig 10

Unit Circle and Trig Values

1 30° 1 2 3 2

Unit Circle and Trig Values Right Triangle Trig

Example 7.4.1: Find the exact value of each expression. a) sin 30 cos 30 sin120 b) cos225 sin315 4 cos60 c) 2 tan

• 2 cos