Chapter 8

# Chapter 8 - Chapter8: (8.1 1 sin s t = sin s cos t cos s...

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Chapter 8: Analytical Trigonometry The addition formulas for sine and cosine function (8.1) Distributive law from algebra will not work to add and subtract angels under trig functions as shown below: 1)sin( s + t ) = sin s cos t + cos s sin t 2)sin( s t ) = sin s cos t cos s sin t 3)cos( s + t ) = cos s cos t sin s sin t

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4)cos( s t ) = cos s cos t + sin s sin t 5)tan( s + t ) = tan s + tan t 1 tan s tan t 6)tan( s t ) = tan s tan t 1 + tan s tan t sine and cosine functions follow the rules shown below to add or subtract angels under them. You must memorize the above identities since they will allow you to successfully add and subtract angels under sine, cosine and tangent functions.
Ex : 1)cos( π + θ ) = 2)sin( θ 3 π 2 ) 3)cos75 0 =

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4)sin105 0 = 5)sin15 0 = 6)sin( π 12 ) = 7)sin( 7 π 12 ) =
Now let’s look at the following two identities under a new light by applying these addition formulas: cos( π 2 α ) = sin α sin( π 2 β ) = cos β Remember, in the previous sections we proved the above identities by looking at it from the graphical perspective. Proof of one of the tan identity: tan( s + t ) =

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Ex: Simplify the following quantities : 1)tan( π 12 ) = 2)tan(75 0 ) =
3) Given sin α = 12 13 , π 2 < α < π and cos β = 3 5 , π < β < 3 π 2 Find sin( α + β ) = cos( α + β ) = sin( α - β ) = cos( α - β ) =

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The Double Angel Formulas (8.2) By using the summation of angels formulas from the previous section, derive the double angel formulas for sin 2 θ , cos2 θ , tan2 θ ; use the fact 2 θ = θ + θ sin 2 θ = cos2 θ = tan 2 θ =
Given sin θ = 3 4 , π 2 < θ < π Find sin2 θ = cos2 θ = tan2 θ =

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The Half‐Angel Formulas: 1)sin( s 2 ) = ± 1 cos s 2 2)cos( s 2 ) = ± 1 + cos s 2 3)tan( s 2 ) = ± 1 cos s 1 + cos s = ± sin s 1 + cos s Proof of 1) just for your amusement:

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Ex : Use sin θ = 3 4 , π 2 < θ < π as before but now find sin( θ 2 ) = cos( θ 2 ) =
tan( θ 2 ) = Ex : Find sin( π 8 ), cos( π 8 ), and tan( π 8 ) using half‐angel formulas:

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