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# Triginometry MTH 181

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This 5 page Class Notes was uploaded by Joseph Notetaker on Thursday March 17, 2016. The Class Notes belongs to MTH 181 at Missouri State University taught by Micheal E. Cagle in Spring 2016. Since its upload, it has received 34 views. For similar materials see Trigonometry in Math at Missouri State University.

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Date Created: 03/17/16

Trigonometry Notes Chapter 2 Section 1&2 Pythagorean Identities: Sin θ+Cos θ=1 Tan θ+1=Sec θ 2 1+Cot θ=Csc θ 2 Conjugates: 2 2 A+B--------A-B so (a+b)*(a-b)=a -b Verify the Identity: sinθ 1+cosθ = Binomial in the denominator is not good 1−cosθ sinθ sinθ ∗(1+cosθ) 1−cosθ Multiply by conjugate (1+cosθ) sinθ(1+cosθ) 2 (1-cosθ)(1+cosθ)=sin2θ because 1-cosθ=sinθ sin θ 1+cosθ 1+cosθ = Factor out sinθ sinθ sinθ Replace with single term: Tan θ+1=Sec θ 2 Sec θ+1=N/A Verify the Identity: tanθ secθ−1 =cscθ+cotθ Left side tanθ ∗(secθ+1) secθ−1 Multiply by the conjugate (secθ+1) tanθ(secθ+1) 2 (secθ1)(secθ+1)=tan θ tan θ secθ+1 tanθ Factor out tanθ 1 cosθ +1 Express as sin and cos sin cos 1 +1 ∗ cosθ Multiply (cosθ )( sinθ 1 cosθ 1 + cosθ over cosθ cancels leaving sinθ sinθ sinθ cscθ+cotθ=cscθ+cotθ Simplify Verify the identity: 2 tanθ+cotθ= tan θ+1 tanθ Move to Left side sinθ+cosθ cosθ sinθ Express as sin and cos sinθ sinθ cosθ cosθ ∗( ) + ∗( ) Multiply by reciprocals cosθ sinθ sinθ cosθ 2 2 sin θ + cos θ iθ oθ siθ cθ Simplify 2 1 = tan θ+1 2 2 iθ oθ tanθ sin θ+cos θ=1 tan θ+1 tanθ Move to Right side sec θ tan θ+1=sec θ tanθ 1 2 cos θ sinθ Express as sin and cos cosθ 1 2 ∗cosθ cos θ Multiply sinθ 1 1 cosθ∗sinθ = sinθ∗cosθ Section 2 sin α−sin α=cos α−cos α 4 4 4 sin α+cos α ≠ 1 2 2 4 2 2 2 4 Instead: sin α=(1−cos α) and sin α=(1−cos α)(1−cos α) or 1−2cos α+cos α Here are some example problems: 1 1 + =−2cscxcotx Move to left side cosx+1 cosx−1 1 cosx−1 1 cosx+1 ∗( )+ ∗ ( ) Multiply by conjugates cosx+1 cosx−1 cosx−1 cosx+1 cosx−1 cosx+1 2 + 2 Now that the denominator matches add the fractions −sin x −sin x 2cos 2 −sin x (cosx+1) + (cosx1) = 2cos −2∗1 ∗cosx sinx Express as the product of fractions sinx −2cscxcotx Simplify 1+secθ sinθ+tanθ =csc Move to left side 1 1+ cosθ Express as sin and cos sinθ+ sinθ cosθ cosθ+1 niθ s+sinθ Multiply all terms by cos cosθ+1 sinθ(cosθ+1) Factor sinθ out of the denominator 1 sinθ Factor out cosθ+1 csc Simplify

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