Trigonometric Formulas and Equations - Unit 4

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Questions and Answers

What is the Pythagorean theorem?

a^2 + b^2 = c^2

Which of the following are trigonometric identities? (Select all that apply)

  • Cos(2x) = Cos^2(x) - Sin^2(x) (correct)
  • Sin^2(x) + Cos^2(x) = 1 (correct)
  • Sin(x + y) = Sin(x) + Sin(y)
  • Tan(x) = Sin(x)/Cos(x) (correct)

The tangent function is an even function.

False (B)

What is the double angle formula for sine?

<p>Sin(2x) = 2Sin(x)Cos(x)</p> Signup and view all the answers

The half angle formula for cosine is Cos(x/2) = ________.

<p>±√((1 + Cos(x))/2)</p> Signup and view all the answers

What is the formula for converting a sum to a product in trigonometry?

<p>Cos(a + b) = Cos(a)Cos(b) - Sin(a)Sin(b) (A), Sin(a + b) = Sin(a)Cos(b) + Cos(a)Sin(b) (C), Sin(a - b) = Sin(a)Cos(b) - Cos(a)Sin(b) (D)</p> Signup and view all the answers

What types of problems can be solved using trigonometric equations?

<p>Finding angles or sides in triangles, modeling periodic phenomena, etc.</p> Signup and view all the answers

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Study Notes

Trigonometric Identities

  • Pythagorean Identity: Relates the squares of sine and cosine functions, expressed as ( \sin^2(x) + \cos^2(x) = 1 ).
  • Even and Odd Functions:
    • Even: ( \cos(-x) = \cos(x) )
    • Odd: ( \sin(-x) = -\sin(x) ) and ( \tan(-x) = -\tan(x) )

Sum and Difference Formulas

  • Used to express trigonometric functions of sums and differences of angles.
  • Sine: ( \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) )
  • Cosine: ( \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) )
  • Tangent: ( \tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a)\tan(b)} )

Double Angle and Half Angle Formulas

  • Double Angle:
    • Sine: ( \sin(2x) = 2\sin(x)\cos(x) )
    • Cosine: ( \cos(2x) = \cos^2(x) - \sin^2(x) ) or ( 2\cos^2(x) - 1 ) or ( 1 - 2\sin^2(x) )
    • Tangent: ( \tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)} )
  • Half Angle:
    • Sine: ( \sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos(x)}{2}} )
    • Cosine: ( \cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos(x)}{2}} )
    • Tangent: ( \tan\left(\frac{x}{2}\right) = \frac{\sin(x)}{1 + \cos(x)} ) or ( \frac{1 - \cos(x)}{\sin(x)} )

Sum-to-Product and Product-to-Sum Formulas

  • Sum-to-Product: Converts sums of sine and cosine functions into products.
    • ( \sin(a) + \sin(b) = 2\sin\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right) )
    • ( \cos(a) + \cos(b) = 2\cos\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right) )
  • Product-to-Sum: Converts products of sine and cosine into sums.
    • ( \sin(a)\sin(b) = \frac{1}{2}[\cos(a-b) - \cos(a+b)] )
    • ( \cos(a)\cos(b) = \frac{1}{2}[\cos(a+b) + \cos(a-b)] )
    • ( \sin(a)\cos(b) = \frac{1}{2}[\sin(a+b) + \sin(a-b)] )

Solving Trigonometric Equations

  • Techniques include using identities, algebraic manipulations, and inverse trigonometric functions.
  • General solutions may involve periodicity, represented as ( x = x_0 + nT ) where ( T ) is the period and ( n ) is any integer.

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