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Exploring Trigonometric Functions: Values, Graphs, and Equations

This guide covers essential aspects of trigonometric functions, including finding specific values using graphs, solving equations for angles, and graphing functions over defined intervals. Key topics include evaluating sine, cosine, tangent, secant, and cosecant at various degrees and radians, identifying amplitude, period, and phase shift, and constructing equations for sine and cosine functions. It is designed for students looking to strengthen their understanding of trigonometry concepts and their applications in problem-solving.

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Exploring Trigonometric Functions: Values, Graphs, and Equations

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  1. Review 6.1-6.4

  2. Find each value by referring to the graphs of the trig functions 0 0 -1 0 Undefined -1 sin (-720°) tan (-180°) cos (540°) tan (π) csc (4π) sec (π)

  3. Find the values of θ for which each equation is true. 270° + 360k° where k is any integer 180° + 360k° where k is any integer 180k° where k is any integer 180k° where k is any integer sin θ = -1 sec θ = -1 tan θ = 0 sin θ = 0

  4. Graph each function on the given interval. 1.) y = sin x [-90° ≤ x ≤ 90°], scale of 45°

  5. Graph each function on the given interval. 1.) y = tan x [-π/2 ≤ x ≤ 3π/2], scale = π/4

  6. Graph each function on the given interval. 1.) y = cosx [-360° ≤ x ≤ 360°]

  7. Graph each function on the given interval. 1.) y = sec x [-360° ≤ x ≤ 360°]

  8. State the amplitude, period, and phase shift for each function. Amp = 2 Per = 360° PS = 0° Amp = 10 Per = 360° PS = 0° Amp = 3 Per = 90° PS = 0° Amp = 0.5 Per = 360° PS = right Amp = 2.5 Per = 360° PS = 180° left Amp = 1.5 Per = 90° PS = right 1.) y = -2sin θ 2.) y = 10sec θ 3.) y = -3sin 4θ 4.) y = 0.5sin (θ- ) 5.) y = 2.5 cos(θ + 180°) 6.) y = -1.5sin (4θ- )

  9. Write an equation of the sine function with each amplitude, period, and phase shift y = ± 0.75 sin(θ- 30°) y = ± 4 sin(120θ+3600°) 1.) Amp = 0.75, period = 360°, PS = 30° 2.) Amp = 4, period = 3°, PS = -30°

  10. Write an equation of the sine function with each amplitude, period, and phase shift y = ± 3.75 cos (4θ- 16°) y = ± 12 cos (8θ- 1440) 1.) Amp = 0.75, period = 360°, PS = 30° 2.) Amp = 4, period = 3°, PS = -30°

  11. Graph each function: 1.) y = 0.5 sin x

  12. Graph each function: 1.) y = 2 cos (3x)

  13. Graph each function: 1.) y = 2 cos (2x – 45°)

  14. Graph each function: 1.) y = tan (x + 60°)

  15. Find the exact value of each expression without using a calculator. When your answer is an angle, express it in radians. Work out the answers yourself before you click.

  16. Answers for problems 1 – 9. y 2 1 x The reference angle is so the answer is Negative ratios for arccos generate angles in Quadrant II.

  17. y 14. 2 x -1 y 15. 1 x 2

  18. Graph each function: 1.) y = -2cos (3θ), scale π/4, -2π ≤ θ ≤ 2π

  19. Graph each function: 1.) y = ½ cos(x – π/2)

  20. Find the values of x in the interval 0 ≤x ≤ 2π that satisfies the equation: 1. 2. 3. 4.

  21. Evaluate each expression. Assume all angles are in Quadrant I 1. 2.

  22. Quick Quiz Find the values of θ for which the equation tan θ = 1 is true. State the domain and range for the function y = -cscx State the amplitude, period, and phase shift of: Write an equation of the cosine function with amplitude 7, period π, and phase shift 3π/2 45°+180°k D = all reals except 180°k R = y ≤ -1 or y ≥ 1 A = ⅓ PS = -π/6 P = 2π

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