Pre-cal Presentation 4 Connor Mclaughlin

Pre-Cal Presentation 4 Connor McLaughlin Page 449: #112 Express each of the other trigonometric functions in terms of sin(x)

Cos(x) 

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To find cosine, you use a Pythagorean identity sin2(x)+cos2(x)=1 Do some re-arranging: 

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Take the square root of both sides 

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cos2(x)=1-sin2(x) √cos2(x)= √1-sin2(x)

Leaves you with 

cos(x)=√1-sin2(x)

Tan(x) 

Use the quotient identity of tangent 

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Substitute our known value of cos(x) 

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Tan(x)=sin(x)/cos(x) Tan(x)=sin(x)/√1-sin2(x)

Because square roots are not allowed to remain on the bottom, you must multiply the top and bottom by √1-sin2(x) 

Tan(x)=[sin(x)√1-sin2 (x)]/1-sin2 (x)

Tan(x) cont. 

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Then square both top and bottom  Tan2 (x)=sin2(x)[1-sin2(x)]/([1-sin2(x)][1-sin2 (x)] Cancel  Tan2(x)=sin2(x)[1-sin2(x)]/([1-sin2(x)][1-sin2 (x)]  Tan2(x)=sin2(x)/1-sin2(x) Make into two fractions  Tan2(x)=[Sin2(x)/1]-[sin2(x)/sin2(x)]  Tan2(x)=sin2(x)-1

Square root of both sides  Tan(x)=√sin2(x)-1

Cot(x) 

Use reciprocal identity of cotangent 

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Cot(x)=1/tan(x)

Insert our known value of tan(x) 

Cot(x)=1/√sin2(x)-1

Csc(x) 

For cosecant, you use the reciprocal identity of sine 

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csc(x)=1/sin(x)

That’s all that there is to that

Sec(x) 

Use reciprocal identity of secant 

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Sec(x)=1/cos(x)

Insert known value of cosine 

Sec(x)=1/√1-sin2(x)

Summary     

Cos(x)=√1-sin2(x) Tan(x)=√sin2(x)-1 Cot(x)=1/√sin2(x)-1 Csc(x)= 1/sin(x) Sec(x)=1/√1-sin2(x)