The Unit Circle - Germanna Community College
The Unit CircleThe unit circle can be used to calculate the trigonometric functions sin(θ), cos(θ), tan(θ), sec(θ), csc(θ),and cot(θ). It utilizes (x,y) coordinates to label the points on the circle, where x represents cos(θ) of a𝑦given angle, y represents sin(θ), and 𝑥 represents tan(θ). Theta, or θ, represents the angle in degrees orradians. This handout will describe unit circle concepts, define degrees and radians, and explain theconversion process between degrees and radians. It will also demonstrate an additional way of solvingunit circle problems called the triangle method.What is the unit circle?The unit circle has a radius of one. The intersection of the x and y-axes (0,0) is known as the origin. Theangles on the unit circle can be in degrees or radians.DegreesDegrees, denoted by , are ameasurement of angle size that isdetermined by dividing a circle into360 equal pieces.RadiansRadians are unit-less but are alwayswritten with respect to π. Theymeasure an angle in relation to asection of the unit circle’scircumference.The circle is divided into 360 degrees starting on the right side of the x–axis and movingcounterclockwise until a full rotation has been completed. In radians, this would be 2π. The unit circleis shown on the next page.Converting Between Degrees and RadiansIn trigonometry, most calculations use radians. Therefore, it is important to know how to convertbetween degrees and radians using the following conversion factors.Conversion Factors𝑫𝒆𝒈𝒓𝒆𝒆𝒔 𝝅 𝑹𝒂𝒅𝒊𝒂𝒏𝒔𝟏𝟖𝟎 𝟏𝟖𝟎 𝑹𝒂𝒅𝒊𝒂𝒏𝒔 𝑫𝒆𝒈𝒓𝒆𝒆𝒔𝝅Example 1:Convert 120 to radians.Step 1: If starting with degrees, 180 should beon the bottom of the conversion factor so thatthe degrees cancel.120 11Provided by the Academic Center for Excellence1𝜋120 (𝜋) 2𝜋 180 1(180 )3The Unit CircleUpdated October 2019
The Standard Unit CircleY - AxisII(0,1) 1 ξ3ቆ , ቇ2 2 ξ2 ξ2ቆ, ቇ22π22π3ξ2 ξ2ቆ , ቇ2 2π33π490 120 ξ3 1ቆ, ቇ2 25π6π460 135 π30 210 ξ3 1ቆ, ቇ22240 ξ2 ξ2ቆ,ቇ22III270 4π3 1 ξ3ቆ ,ቇ223π2(0, 1)(1,0)300 ξ3 1ቆ , ቇ2 27π45π3ξ2 ξ2ቆ ,ቇ221 ξ3ቆ ,ቇ2 2Key: (𝐂𝐨𝐬(𝛉), 𝐒𝐢𝐧(𝛉))𝑻𝒂𝒏(𝜽) 𝐒𝐢𝐧(𝛉)𝐂𝐨𝐬(𝛉)22Provided by the Academic Center for Excellence2X - Axis11π6315 225 5π40π2π0 360 330 180 7π6ξ3 1ቆ , ቇ2 2π645 150 ( 1,0)I1 ξ3ቆ , ቇ2 2The Unit CircleUpdated October 2019IV
The Unit Circle by TrianglesAnother method for solving trigonometric functions is the triangle method. To do this, the unit circle isbroken up into more common triangles: the 45 45 90 and 30 60 90 triangles. Some examples ofhow these triangles can be drawn are below.30 60 90 Triangle45 45 90 Triangleξ2ξ3Sides:Angles:1 45 1 45 Sides:ξ2 90 Angles:1 30 ξ3 60 2 90 Triangle Method Steps1. Choose a triangle. If the angle inside the trigonometric functionis divisible by 45, use the 45 45 90 triangle. IIIIIIIVIf the angle is divisible by 30 or 60, usethe 30 60 90 triangle.2. Draw the triangle in the correct quadrant, with thehypotenuse pointed towards the origin. Add negative signs on the sides if necessary.3. Analyze the triangle.4. Rationalize and simplify.33Provided by the Academic Center for Excellence3The Unit CircleUpdated October 2019
Example 2:Use the triangle method to solve:𝐶𝑜𝑠(45 )Step 1: Choose a triangle.Because 45 is divisible by 45, use the 45 45 90 triangle.Step 2: Draw the triangle in the correct quadrant.This triangle will be in quadrant I because 45 is between 0 and 90 .ξ2Step 3: Analyze the triangle.𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡Remember that cos(θ) represents 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒. Here, the adjacent side to θ (or 45 ) is 1,and the hypotenuse is ξ2. This results in 𝑐𝑜𝑠(45 ) 1.ξ2Step 4: Rationalize the denominator.The denominator is rationalized by removing the square roots. Do this by multiplyingthe numerator and denominator of the resulting fraction1ξ2by the radical in thedenominator ξ2.Cos(45 ) Cos(45 ) 1ξ2 ξ2ξ2ξ2244Provided by the Academic Center for Excellence4The Unit CircleUpdated October 2019
Example 3:Use the triangle method to solve:𝑇𝑎𝑛(240 )Step 1: Choose a triangle.Because 240 is divisible by 30, use the 30 60 90 triangle.Step 2: Draw the triangle in the correct quadrant.This triangle will be in quadrant III because 240 is between 180 and 270 . Additionally,60 will be the angle near the origin because 240 is 60 more than 180 . ξ3Step 3: Analyze the triangle.𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒Note that tan(θ) represents 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 . Here, the opposite side is ξ3 while the adjacentside is 1. This results in 𝑡𝑎𝑛(240 ) ξ3 1.Step 4: Simplify.The negatives cancel each other out to leaveξ3,1which is ξ3.𝑇𝑎𝑛(240 ) ξ355Provided by the Academic Center for Excellence5The Unit CircleUpdated October 2019
Practice Problems:Find the exact value of the problems below using either the standard unit circle or the triangle method.1.) Sin4𝜋32.) Cos11𝜋6𝜋3.) Tan 34.) Cos5.) Sin 2𝜋3(Hint: Instead of rotating counterclockwise around the circle, go clockwise.) 𝜋26.) Tan 2𝜋𝜋7.) Tan 29𝜋8.) Cos 4 (Hint: For angles larger than 360 , continue going around the circle.)Answers:1.)2.) ξ32ξ323.) ξ34.) 125.) 16.) 019.) Undefined (0 cannot occur/does not exist)7.)ξ2266Provided by the Academic Center for Excellence6The Unit CircleUpdated October 2019
unit circle problems called the triangle method. What is the unit circle? The unit circle has a radius of one. The intersection of the x and y-axes (0,0) is known as the origin. The angles on the unit circle can be in degrees or radians. The circle is divided into 360 File Size: 442KB
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