Trigonometric FunctionsPre-Calculus

Amanda Woodbury

Our Friend the Unit Circle!

radius=1

Ɵ

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This circle will be used for everything in this section. It helps us with all the functions, ratios, and calculations we will learn. This is why it is our new best friend.

Using the Unit CircleTrigonometry

• sin Ɵ =

• cos Ɵ =

• tan Ɵ = =

• csc Ɵ = =

• sec Ɵ = =

• cot Ɵ = =

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sin Ɵ cos Ɵ

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cos Ɵsin Ɵ

1 1sin Ɵ

1 1cos Ɵ

csc=cosecant sec=secant cot=cotangent

Graphing Using the Unit Circle

Steps to Graphing on the Unit Circle

1. When given an angle (ex. 135°), draw a curve from 0° to the given angle.

90°

180° 0/360°

240°

2. Draw a line connecting the curve you just drew to the edge of the circle.

3. Draw a dashed line from the edge of the circle to make a right triangle.

4. Calculate the angle measure inside the triangle (180-135 = 45)

5. Find the sine, cosine, and tangent of the angle (you may use a calculator for this if you wish).

90°

180° 0/360°

240°

45°

Trigonometry

sin Ɵ = = 3/4

cos Ɵ = = 3/5

tan Ɵ = = 4/3

3

45

Ɵ

opphypadjhyp

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Using the triangle at right, solve for sin Ɵ, cos Ɵ, and tan Ɵ.

Using the triangle at right, solve for csc Ɵ, sec Ɵ, and cot Ɵ.

csc Ɵ = = 5/4

sec Ɵ = = 5/3

cot Ɵ = = 3/4

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hypadj

adjopp

3

45

Ɵ

Radian MeasureRadians use π ratios instead of degrees and are

fractions, not whole numbers , π, 2π, etc…π , π , π , π , 3π , 5π

6 4 3 2 4 6

Degree MeasureDegrees do not use π ratios and are whole numbers

instead of fractions15°, 30°, 45°, 60°, 90°, 120°, 180°, 360°, etc…

90°

180° 0/360°

240°

Converting between Radians and Degrees

When converting from Radians to Degrees:1. Multiply the radian ratio by 180

2. Divide the radian ratio by π

3. The two π symbols will cancel each other out and all that will be left is a simple mathematical equation

Example:

= = 150°5π 6

180 π

5(180) 6

Converting Between Degrees and Radians

When converting from Degrees to Radians:1. Multiply the degree measure by π

2. Divide the degree measure by 180

Example:

120° π = 120π = 2π___180

____ 180

__ 3

Arc Length of a Given AngleThe equation for finding the arc length of a given angle is:

s = dπƟ

s = arc length

d = diameter (2r)

Ɵ = given angle

____ 360

Ɵ

s

r

Arc Length of an Angle in a Given CircleFind the arc length of 60° in a circle with radius = 5.

r=5

60°

s = dπƟ 360s = 10π60 360s = 600π 360s = 5π 3s = 5.236

CitationsInformation:

http://www.math-prof.com/Geom/Geom_Ch_32.aspAll other information is from my previous pre-calculus

experience. I am a math major, so I should know all of this anyway.

Citations cont…Pictures:

Title slide pictures, photos on slides 10 & 11, and teacher photo on Information Citation slide from PowerPoint Clip Art

Unit Circle and Triangles made by me using PowerPointPi Pie photo on slide 10 from: pauladamsmith, 13 January

2008 via Flickr, Creative Commons Attribution.

Citations cont…Lesson Plan from: http://www.michigan.gov/documents/

PreCalc_167750_7.pdfLesson P6.1 – “Define (using the unit circle), graph, and use

all trigonometric functions of any angle. Convert between radian and degree measure. Calculate arc lengths in given circles.”