Assignment on The Unit Circle: Simple Trigonometry - Precalculus | MAT 170 | Assignments Pre-Calculus

Name:____________________

TheUnitCircle-SimpleTrigonometry.



Trigonometryisthestudyofanglesandthephysicalrelationshipsbetweenanglesandgeometry.Tostart,

weusetheunitcircle,whichisacircleofradius1unit,centeredattheorigin.



Part1:Findingcoordinatesofpointsontheunitcircle.



Fourofthecoordinatesareeasy:(1,0),(0,1),(-1,0)and(0,-1).Theselieonthexan dyaxesandarecalled

cardinal points (i.e. like the points on a compass).Drawanyray fromth eorigin intersecting the circle.

CalltheintersectionpointP.Thisrayhasanangleoftheta"θ".Therefore,thecoordinatesofpointParea

functionoftheangleθ.Wemakethefollowingdefinition:



GivenpointPontheunitcircle.PointPhascoordinates(x,y).



x=cosθ

θθ

θ,andy=sinθ

θθ

θ.



Simplegeometry(righttrianglesandpythagoras' formula)willh elplocatevaluesforsomean glessuchas

30°,45°and60°,whileyourcalculatorwillassistforotherangles.



Firstquadrantvaluesforcosθandsinθ.Thatmeans0<θ<90(degrees)or0<θ<π/2(radians).



E D



     C



     B



      A



Fillinthefollowingtablebasedonthefigureabove:



Angleθindegrees Angleθinradian s x=cosθy=sinθ

A.0   

B.30   

C.45   

D.60   

E.90   



Watchyourcalculatorsettings!



Questions:



1. Whatisthedistan cefrompointBtotheorigin?Why?



2. Arayhasanan gleofθ=20°.Findthepoint'scoordinates.



3. WhatwouldthecoordinatesofPbeforanangleof390°?Why?



Partial preview of the text

Download Assignment on The Unit Circle: Simple Trigonometry - Precalculus | MAT 170 and more Assignments Pre-Calculus in PDF only on Docsity!

The Unit Circle - Simple Trigonometry. Trigonometry is the study of angles and the physical relationships between angles and geometry. To start,we use the unit circle, which is a circle of radius 1 unit, centered at the origin.

Part 1: Finding coordinates of points on the unit circle. Four of the coordinates are easy: (1,0), (0,1), (-1,0) and (0,-1). These lie on the x and y axes and are called cardinal points ( i.e. like the points on a compass). Draw any ray from the origin intersecting the circle. Call the intersection point P. This ray has an angle of theta "θ". Therefore, the coordinates of point P are a function of the angle θ. We make the following definition: Given point P on the unit circle. Point P has coordinates (x,y). x = cos θθθθ , and y = sin θθθθ. Simple geometry (right triangles and pythagoras' formula) will help locate values for some angles such as 30 °, 45° and 60°, while your calculator will assist for other angles. First quadrant values for cos θ and sin θ. That means 0 < θ < 90 (degrees) or 0 < θ < π/2 (radians).

E D C B A Fill in the following table based on the figure above:

Angle A. 0 θ in degrees Angle θ in radians x = cos θ y = sin θ B. 30 C. 45 D. 60 E. 90 Watch your calculator settings! Questions:

What is the distance from point B to the origin? Why?
A ray has an angle of θ = 20°. Find the point's coordinates.
What would the coordinates of P be for an angle of 390°? Why?

Part II: Values of sin θθθθ and cos θθθθ for the other quadrants. Use your calculator to complete the following table. Some patterns should be evident. Angle in degrees Angle in radians Which quadrant? (^) x = cos θ y = sin θ 120 135 150 210 225 240 300 315 330 Questions:

In quadrant II, the cos θ is always _________, and the sin θ is always __________.
In quadrant III, the cos θ is always _________, and the sin θ is always __________.
In quadrant IV, the cos θ is always _________, and the sin θ is always __________.
Look at your results for the angles 135°, 225° and 315°, and compare them to the results for the angle of 45° from Quadrant I. What do you notice?

(The angle 45° is called the reference angle for the angles 135°, 225° and 315°.

What is the reference angle for 120°?
What is the reference angle for 335°?
In general, if an angle θ is in the second quadrant, how would you find its reference angle?
In general, if an angle θ is in the third quadrant, how would you find its reference angle?
In general, if an angle θ is in the fourth quadrant, how would you find its reference angle?

Part III: The tangent function (tan (^) θθθθ ). The tangent of an angle is equivalent to the slope of the ray. In terms of sin θ and cos θ, we can define tan θ = (sin θ)/(cos θ). Fill in the following table: Angle in degrees (^) Tan θ 0 30 45 90

Can you explain why the tangent at 90° is undefined?

Assignment on The Unit Circle: Simple Trigonometry - Precalculus | MAT 170, Assignments of Pre-Calculus

Related documents

Partial preview of the text

Download Assignment on The Unit Circle: Simple Trigonometry - Precalculus | MAT 170 and more Assignments Pre-Calculus in PDF only on Docsity!