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If θ = 9π/4, then find exact values for the following:

sin θ
cos θ
tan θ

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Final answer:

For θ = 9π/4, the equivalent angle within [0, 2π) is π/4. Therefore, the exact values are sin(θ) = √2/2, cos(θ) = √2/2, and tan(θ) = 1, since these are the trigonometric values for an angle of π/4.

Step-by-step explanation:

If θ = 9π/4, we first reduce this angle by subtracting multiples of 2π (the full circle in radians), until the angle is within the range [0, 2π). Since 9π/4 is equivalent to 2π + π/4, the angle co-terminal with 9π/4 that lies within the desired range is θ = π/4.

The exact trigonometric values for π/4 are known:

  • sin(π/4) = √2/2,
  • cos(π/4) = √2/2, and
  • tan(π/4) = 1.

Thus, for θ = 9π/4:

  • sin(θ) = sin(9π/4) = sin(π/4) = √2/2,
  • cos(θ) = cos(9π/4) = cos(π/4) = √2/2,
  • tan(θ) = tan(9π/4) = tan(π/4) = 1.

Note: These values are the same as for θ = π/4 because trigonometric functions are periodic, and they repeat every 2π radians.

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