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Maths MethodsTrigonometric equations K McMullen 2012
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The Unit CircleTo help us understand the unit circle let’s firstlook at the right-angled triangle with ahypotenuse of 1 unit in length. The reason why itis 1 unit is because we consider thecircumference of the circle to be 2π(circumference of a circle is 2πr, therefore r=1which is the hypotenuse) this will be explained inmore detail later. K McMullen 2012
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The Unit CircleUsing the cosine ratio in the triangle:cosϑ=a/1a=cosϑUsing the sine ratio in the triangle:sinϑ=b/1b=sinϑThis means that in a unit circle the horizontal length in the measureof the cosine of the angle and the vertical length is a measure of thesine of the angle. K McMullen 2012
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The Unit CircleWhen this triangle is places inside a circle, it canbe used to find the trig ratios shown preiviously.The value of cosϑ can be read off the x-axis.The value of sinϑ can be read off the y-axis.The value of tanϑ can be read off a verticaltangent line drawn on the right side of theunit circle.A copy of the unit circle is in the next slide K McMullen 2012
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Converting between radians and degreesAngles are measured in degrees or radians.To define a radian we can use a circle which has aradius of one unit. This circle is called the unit circle.When working with degrees we know that onerevolution of a circle is 360°. When working withradians one revolution is 2π (this is because we areworking with a circle with a radius of 1).Therefore: 360°=2π 180°=π K McMullen 2012
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Converting between radians and degreesThe radius of the circle can be any length and can still beregarded as a unit. As long as the arc is the same lengthas the radius, the angle will always measure one radian.An angle of 1 degree can be denoted as 1°.A radian angle can be denoted as 1c but we usually leaveoff the radian sign.Therefore, 1c=180°π 1°=π180 K McMullen 2012
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Converting between radians and degreesAlways make sure your calculator is in radianmode when working with radians and degreemode when working with degrees.When working with radians and the unit circle weare no longer referring to North, East, South andWest like we would with a compass. With the unitcircle we use a set of axes (the Cartesian plane)with the x-axis as the horizontal and the y-axis asthe vertical. Remember that cosϑ=x and sinϑ=y. K McMullen 2012
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Converting between radians and degreesBelow is a copy of the unit circle. You need tofamiliarize yourself with the values from this unitcircle so make sure you remember the table ofexact values that we did previously (it’s easy torecalculate these if needed by simply redrawingthe two triangles). K McMullen 2012
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Exact Values Using the equilateral triangle (of side length 2 units), the following exact values can be found:(look at page 271 to get the exact values- these go to the right of each ‘=‘sign) sin30°=sinπ/6= sin60°=sinπ/3= cos30°=cosπ/6= cos60°=cosπ/3= tan30°=tanπ/6= tan60°=tanπ/3= K McMullen 2012
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Exact ValuesUsing the right isosceles triangle with two sidesof length 1 unit, the following exact values can befound:sin45°=sinπ/4=cos45°=cosπ/4=tan45°=tanπ/4= K McMullen 2012
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Symmetry Formulae The unit circle is symmetrical so that the magnitude of sine, cosine and tangent at the angles shown are the same in each quadrant but the sign varies.We’ll go over this in more detail in class K McMullen 2012
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Trigonometric IdentitiesWhen a right-angled triangle is placed in the firstquadrant of a unit circle, the horizontal side has thelength of cosϑ and the vertical side has the length ofsinϑ.Therefore, using the tan ratio(tanϑ=opposite/adjacent): tanϑ=sinϑ/cosϑUsing Pythagoras’ theorem (a2+b2=c2): (sinϑ)2+(cosϑ)2=12 sin2ϑ+cos2ϑ=1 K McMullen 2012
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Complementary AnglesComplementary angles add to 90° or π/2 radians.Therefore, 30° and 60° are complementary angles. Inother words π/6 and π/3 are complementaryangles, and θ and π/2-θ are also complementaryangles.The sine of an angle is equal to the cosine of itscomplement. Therefore, sin60°=cos(30°). We say thatsine and cosine are complementary functions.The complement of the tangent of an angle is thecotangent or cot- that is, tangent and cotangent arecomplementary functions (as well as reciprocalfunctions). K McMullen 2012
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Complementary AnglesCopy the table from page 270 into your notes K McMullen 2012
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