TRIGONOMETRY Math 102

COURSE DESCRIPTION• A course to teach students the basic concepts of trigonometry like evaluation of the six trigonometric functions, derivation of fundamental trigonometric identities, proving of simple trigonometric identities, etc. It also includes solutions involving the right triangle and utilization of the laws of sines and cosines. Students are also expected to sketch and discuss the behavior of the different curves involving trigonometric functions.

HIPPARCHUS OF NICEA• Father of Trigonometry• Able to approximate the radius of the earth by using the concept of trigonometry

• r = 4, 080 miles

TRIGONOMETRY•Greek words: Trigonon – “Triangle” and metron – “Measure”

•Development of knowledge in the field of architecture, navigation, survey and astronomy

TRIGONOMETRY•Dealing with triangles and angle measurements

•Trigonometric ratios and identities

ANGLE MEASUREMENT

Math 102 - Trigonometry

ANGLES

Initial side

terminal side

θ

Standard Position •The vertex is at the origin and its initial side on the positive x-axis.

•The terminal side falls in on one of the four quadrants

Coterminal Angle •Terminal sides of two angles in standard position, coincide

•θ (n•360)

Find the coterminal angle of the following:

1. -122°2. 470°3. 720°4. -330°5. -270°

ANGLE MEASUREMENTS

•Degree•Radian•Revolution•Degree, minutes, seconds

RADIAN MEASURE• Central Angle – vertex is at the origin and sides are the radii of the circle

• When the central angle intercepts an arc that had the same length as the radius of the circle, the measure of this angle is defined as ONE RADIAN

Degree ↔ Radian

1 °= 𝜋180 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 1𝑟𝑎𝑑𝑖𝑎𝑛=( 180

𝜋 )o

Degree ↔ Radian Convert the following

1. 120°2. -245°3. 4.

Degree ↔ Revolution

3 60 °=1 revolution

Degree ↔ Revolution Convert the following

1. ¼ clockwise revolution

2. ¾ counterclockwise revolution

Degree, Minutes, Seconds

1 = left ({1} over {60} right ) 1 = left ({1} over {3600} right )   o

o

Degree, Minutes, Seconds

Example:Convert the following to DMS and vice versa1. 12.464° 2. 23.42°3. 23°42’45”

ProblemsConvert the following and identify the quadrant where the terminal side lies

Additional ExercisesConvert the following in Degree, Min, Sec

Additional ExercisesConvert the following to radian measure

2. 29

3. 606’20”

TRIGONOMETRY Arc length

Arc length (s)measure of the distance along the curved line making up the arc

s = r • Where: r – radius

- angle measure in radians

Arc length (s)Example:

Find, in centimeters, the length of an arc intercepted by a central angle of 4 radians in a circle with a radius of 3.5 cm.

Arc length (s)Example:

A sector has a radius of 12 cm. and an angle of 65º.  To the nearest tenth of a cm., find its arc length.

Arc length (s)In a circle, a central angle of  1/3 radians subtends an arc of 3 centimeters.  Find the length, in centimeters, of the radius of the circle.

Arc length (s)Skateboarding has become a popular sport. The parks department is thinking of constructing ramps in some of the local playgrounds. A “half-pipe” ramp is formed by two quarter-circle ramps, each of which is 10 feet high, plus a flat space 20 feet long between the centers. Find the distance a skater travels from the top of one ramp to the top of the other

Arc length (s)A 40 -inch pendulum swings through an angle of 18°. Find the length of the arc in inches through which the tip of the pendulum swings.

Arc length (s)The outer diameter of the wheels on a bicycle is 22 inches. If the wheels are turning at a rate of 240 rpm. Find the linear speed of the bike in inches per minute

Arc length (s)A motorcycle is traveling on a curve along a highway. The curve is an arc of a circle with radius of ¼ miles. If the motorcycle’s speed is 42 mph, what is the angle in radians throughwhich the motorcycle will turn in ½ minute.