Chapter 7 analytic trigonometry
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Chapter 7 – Analytic Trigonometry. Section 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas. Double Angle Formulas. The double-angle formulas allow us to find the values of the trigonometric functions at 2 x from their values at x. Proofs of Double Angle Formulas.

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Chapter 7 analytic trigonometry

Chapter 7 – Analytic Trigonometry

Section 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas

7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas


Double angle formulas
Double Angle Formulas

  • The double-angle formulas allow us to find the values of the trigonometric functions at 2x from their values at x.

7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas


Proofs of double angle formulas
Proofs of Double Angle Formulas

Now, you try the proof for cos2 and tan 2.

7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas


Example
Example

  • If and  lies in quadrant II, find the exact value of each of the following:

7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas


Example1
Example

  • Find the exact value of

7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas


Power reducing formulas
Power Reducing Formulas

  • The power reducing formulas play an important part in calculus.

7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas


Half angle formulas
Half Angle Formulas

  • NOTE: The choice of the + or – sign depends on the quadrant in which u/2 lies.

7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas


Example2
Example

  • Find the exact value of cos105o.

7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas


Example3
Example

Verify the identity:

7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas


Example4
Example

Verify the identity:

7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas


Product to sum formulas
Product-to-Sum Formulas

7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas


Examples pg 515
Examples – pg. 515

  • Write the product as a sum.

7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas


Sum to product formulas
Sum-to-Product Formulas

7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas


Examples pg 5151
Examples – pg. 515

  • Write the product as a sum.

7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas


Example pg 515
Example – pg. 515

  • Prove the identity.

7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas


Example5
Example

  • Write an equivalent expression for cos4x that does not contain powers of trigonometric functions greater than 1.

7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas


More examples
More Examples

  • Prove the identity.

7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas


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