Trigonometry - Graphs of Trigonometric Functions - Quick Guides | Mathematics

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Study GuideTrigonometry–Graphs of Trigonometric Functions1. Circular Functions1.1The Unit CircleFigure 1 Unit circle reference.Figure 2 Range of values of trig functions.

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Study GuideThe equationdraws acircleon the coordinate plane. This circle is called theunit circle.•Itscenteris at the origin(0,0).•Itsradiusis 1 unit.The unit circle is extremely important because it helps us understandtrigonometric and circularfunctionsin a visual way.1.2From Angles to NumbersIn regular trigonometry, functions like sine and cosine takeanglesas inputs. Their outputs arerealnumbers.Circular functions work a little differently:•Theirinputsare real numbers (measured inradians).•These numbers representarc lengthsalong the unit circle.•Theiroutputsare still real numbers.They are calledcircular functionsbecause the values come from moving around acircle.1.3How Radians Fit InRadians connect angles to circles naturally.•The radian measure of an angle is based onarc length, not degrees.•Because arc length depends on the circle, radians are perfect for circular functions.•On theunit circle, arc length and angle measure match exactly.This is why trigonometric functions defined on the unit circle lead directly to circular functions.

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Study Guide1.4Defining Sine and Cosine Using the Unit CircleStart with the unit circle (x2+ y2= 1).1.Begin at pointA(1, 0)on the positive x-axis.2.Let (q) be any real number.3.Move|q| unitsalong the circle:oCounterclockwise if (q > 0)oClockwise if (q < 0)4.You arrive at a point (P(x, y).Now define:•sin q = y•cos q = xSo,sine is the y-coordinateandcosine is the x-coordinateof point (P).1.5Other Circular (Trigonometric) FunctionsOnce sine and cosine are defined, the other functions are built from them:These definitions match what you learned in trigonometry—just written in terms of circular functions.1.6Domain of Circular FunctionsBecause you can move any distance around the unit circle:•Thedomainof sine and cosine isall real numbers.•Positive values move counterclockwise.•Negative values move clockwise.So, (sin q) and (cos q) exist forevery real number (q).

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Study Guide1.7Range of Circular FunctionsWhile the domain is unlimited, therangeis not.•Sine and cosine are coordinates on the unit circle.•All x-and y-values on the unit circle lie between −1 and 1.Therefore:This means therangeof both sine and cosine is from −1 to 1.1.8Why Circular Functions MatterCircular functions:•Connect geometry and algebra•Explain why trigonometric values repeat•Make radians feel natural and meaningful•Are essential for advanced math, physics, and engineeringUnderstanding them through the unit circle makes everything clearer and more intuitive.Example1:When Does the Sine Value Equal 1?Question:Which values of (x), between (-2π)and (2π),make the sine function equal to 1?In other words, we are looking for all values of (x) in this interval such that:

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Study GuideFigure 3 Drawing for Example 1.Step 1: Think About the Unit CircleOn the unit circle, the sine of an angle corresponds to they-coordinateof a point on the circle.So asking when (sin x = 1) is the same as asking:When is the point on the unit circle at a height of1?Step 2: Identify Where Sine Equals 1The y-coordinate is equal to 1 only at thetop of the unit circle, which is the point:This means we want all values of (x) that land us at this point.Step 3: Find the Correct Angle MeasuresStarting from(1, 0)and moving around the unit circle:•The first time we reach(0, 1)is at

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Study Guide•If we continue moving around the circle clockwise, we reach the same point again atBoth of these values lie between (-2π)and (2π).Step 4: State the Final AnswerThe values of (x) for which the sine function has a value of 1 in the given interval are:Example2:When Does the Cosine Value Equal −1?Question:Which values of (x), between (-2π)and (2π),make the cosine function equal to (-1)?In mathematical terms, we are solving:Figure 4 Drawing for Example 2.

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Study GuideStep 1: Connect Cosine to the Unit CircleOn the unit circle, the cosine of an angle corresponds to thex-coordinateof a point on the circle.So asking when (cos x =-1) is the same as asking:When is the point on the unit circle all the way to theleft, at an x-value of (-1)?Step 2: Identify the Key PointThe x-coordinate equals (-1) at exactly one point on the unit circle:This is theleftmost pointon the circle.Step 3: Find the Angle ValuesStarting from(1, 0)and moving around the unit circle:•Moving counterclockwise, you reach(-1, 0)at•Moving clockwise, you reach the same point atBoth values are within the interval (-2π ≤x ≤ 2π).Step 4: State the Final AnswerThe values of (x) for which the cosine function has a value of (-1) in the given interval are:

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Study GuideExample3:Finding All Six Circular FunctionsGiven:The pointlies on theunit circle. The arc length from point (A(1,0)to point (P) is (q) units.Question:What are the values of thesix circular (trigonometric) functionsof (q)?Step 1: Use the Coordinates of the PointOn the unit circle:•Thex-coordinategives the cosine value•They-coordinategives the sine valueSo, directly from point (P):Step 2: Find TangentTangent is defined as sine divided by cosine:Substitute the values:

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Study GuideStep 3: Find CotangentCotangent is the reciprocal of tangent (or cosine divided by sine):Step 4: Find SecantSecant is the reciprocal of cosine:Step 5: Find CosecantCosecant is the reciprocal of sine:Step 6: Summary of All Six Values

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Study GuideStep 7: Understanding the SignsThesignsof the trigonometric functions depend on whichquadrantthe point lies in.Summary•Sine represents thevertical positionon the unit circle.•The maximum value of sine is1.•This value occurs whenever the angle places the point at thetop of the circle.•Cosine represents thehorizontal positionon the unit circle.•The minimum value of cosine is−1.

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