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Use the definition or identities to find the exact value of each of the remaining five trigonometric functions of the acute angle θ.

sin θ = √2/6

cos θ =???
tan θ =???
cot θ =???
csc θ =???
sec θ =???

User Dmitry
by
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1 Answer

1 vote

Answer: Given, sin θ = √2/6

We know that sin^2θ + cos^2θ = 1

Solving for cos θ, we get:

cos θ = ±√(1 - sin^2θ)

cos θ = ±√(1 - (2/6)^2)

cos θ = ±√(1 - 1/9)

cos θ = ±√(8/9)

Since θ is acute, cos θ is positive. Hence,

cos θ = √(8/9) = (2√2)/3

We know that tan θ = sin θ / cos θ

tan θ = (√2/6) / [(2√2)/3]

tan θ = √2/4

We know that cot θ = 1/tan θ

cot θ = 1/ (√2/4) = (2√2)/2 = √2

We know that csc θ = 1/ sin θ

csc θ = 1/ (√2/6) = (6√2)/2 = 3√2

We know that sec θ = 1/ cos θ

sec θ = 1/[(2√2)/3] = (3√2)/2

Hence, the exact values of the remaining trigonometric functions of the angle θ are:

cos θ = (2√2)/3

tan θ = √2/4

cot θ = √2

csc θ = 3√2

sec θ = (3√2)/2

User Goodolddays
by
7.9k points
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