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Trigonometric FunctionsPre-Calculus
Amanda Woodbury
Our Friend the Unit Circle!
radius=1
Ɵ
hyp
opp
adj
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 = =
opphypadjhypoppadj
sin Ɵ cos Ɵ
hypopphypadjadjopp
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
oppadj
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
hypopp
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.”