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Understanding Double Angle Identities in Trigonometry

In this section, we explore double angle identities for sine and cosine functions. Learn how to manipulate angles and calculate sine, cosine, and tangent for doubled angles. The identities are: sin(2a) = 2sin(a)cos(a) and cos(2a) = cos²(a) - sin²(a). We also discuss using these identities to prove other trigonometric identities and solve problems. Examples help clarify these concepts, including determining values from known sine and cosine ratios. Get ready to enhance your understanding of trigonometry!

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Understanding Double Angle Identities in Trigonometry

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  1. 7.3.1 – Product/Sum Identities

  2. So far, we have talked about modifying angles in terms of addition and subtraction • Not included within that was the case of multiply certain angles by values • Specifically, what if we double it?

  3. Double-Angle Identities • Double Angle identities will allow us to find trig values for when we double the angle of interest

  4. Sine/Cosine • Let a be an angle; radians or degrees • sin(2a) = 2sin(a)cos(a) • cos(2a) = cos2(a) – sin2(a) = 2cos2(a) – 1 = 1 – 2sin2(a)

  5. Tangent • Tan(2a) = • Just as before, we typically we try to use angles from the unit circle we know about (from our chart)

  6. Similar to problems from the last section, we must be able to use the given identities with or without an angle

  7. Example. Given that sin(x) = 1/√5, and tan(x) is positive, determine the value of cos(2x), sin(2x) and tan(2x)

  8. Example. Given that cos(x) = -2/√5 and that sin(x) is positive, determine the values for cos(2x), sin(2x) and tan(2x).

  9. Proving Identities • Also using product identities, we may verify or prove other identities • Still may need to use previous identities (have those handy, or use the reference page from the back of the book)

  10. Example. Show sin(2x) =

  11. Half-Angle Identities

  12. Using these identities, we can rewrite angles, similar to before • Example. Determine the exact value of sin(π/8) • What angle is π/8 half of?

  13. Example. Determine the exact value for tan(7π/12)

  14. Assignment • Pg. 576 • 1, 3, 5, 12, 19, 20, 22, 24

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