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Notes 4-4

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Basic Identities Involving Sines, Cosines, and Tangents

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Notes 4-4

  1. 1. Section 4-4 Basic Identities Involving Sines, Cosines, and Tangents
  2. 2. Identity An equation that is true for all possible values of the variable
  3. 3. Example 1 Complete the following in your calculator. ⎛ 3π ⎞ 2 ⎛ 3π ⎞ sin ⎜ ⎟ + cos ⎜ ⎟ 2 cos 30° + sin 30° 2 2 ⎝4⎠ ⎝4⎠ 1 1 sin 2 ( −25° ) + cos 2 ( −25° ) cos 2 ( 4π ) + sin 2 ( 4π ) 1 1
  4. 4. Pythagorean Identity For all theta, cos (θ ) + sin (θ ) = 1 2 2
  5. 5. Example 2 If sinθ = , find cosθ. 1 3 sin θ + cos θ = 1 2 2 () 2 + cos θ = 1 2 1 3 − − 1 1 9 9 cos θ = 2 8 9 cos θ = ± 2 8 9 cosθ = ± 8 3 cosθ = ± 22 3
  6. 6. Opposites Theorem For all theta, () cos −θ = cosθ sin ( −θ ) = − sinθ tan ( −θ ) = − tanθ
  7. 7. Example 3 ⎛ π⎞ ⎛π⎞ 2 3 ( ) b.sin ⎜ − ⎟ = − . Find − sin ⎜ ⎟ a.cos30° = . Find cos −30° ⎝ 4⎠ ⎝ 4⎠ 2 2 2 3 − 2 2
  8. 8. Supplements Theorem For all theta in radians, () sin π − θ = sinθ cos (π − θ ) = − cosθ tan (π − θ ) = − tanθ
  9. 9. Complements Theorem For all theta in radians, ⎛π ⎞ sin ⎜ − θ ⎟ = cosθ ⎝2 ⎠ ⎛π ⎞ cos ⎜ − θ ⎟ = sinθ ⎝2 ⎠
  10. 10. Example 4 () ( ) If sin x = .681, find sin -x and sin π - x . () sin -x = −.681 sin (π − x ) = .681
  11. 11. Half-turn Theorem For all theta in radians, () cos π + θ = − cosθ sin (π + θ ) = − sinθ tan (π + θ ) = tanθ
  12. 12. Example 5 Using the unit circle, explain why sin (π − θ ) = sinθ for all θ . On the unit circle, π = 180° . When you measure theta, you start at 0°. So, you’re beginning at points that are reflections of each other. As you plot the values, you will notice they remain as reflections over the y-axis, which will keep the y-coordinates the same, which is sinθ .
  13. 13. Homework p. 256 #1 - 24

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