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If cos ϕ = 0.4626 and 3π/2 < ϕ < 2π, find decimal approximations for sin ϕ and tan ϕ

User MarkK
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Answer: We can use the identity sin^2 ϕ + cos^2 ϕ = 1 to find sin ϕ:

  • cos^2 ϕ + sin^2 ϕ = 1
  • sin^2 ϕ = 1 - cos^2 ϕ
  • sin ϕ = ±√(1 - cos^2 ϕ)

Since 3π/2 < ϕ < 2π, we know that sin ϕ < 0. Therefore, we can take the negative square root:

  • sin ϕ = -√(1 - cos^2 ϕ)
  • sin ϕ = -√(1 - 0.4626^2)
  • sin ϕ ≈ -0.8865

To find tan ϕ, we can use the identity tan ϕ = sin ϕ / cos ϕ:

  • tan ϕ = sin ϕ / cos ϕ
  • tan ϕ = -0.8865 / 0.8861
  • tan ϕ ≈ -0.9996

Therefore, the decimal approximations for sin ϕ and tan ϕ are approximately -0.8865 and -0.9996, respectively.

User Donm
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