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Questions and Answers
What is the Pythagorean theorem?
What is the Pythagorean theorem?
a^2 + b^2 = c^2
Which of the following are trigonometric identities? (Select all that apply)
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.
The tangent function is an even function.
False (B)
What is the double angle formula for sine?
What is the double angle formula for sine?
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The half angle formula for cosine is Cos(x/2) = ________.
The half angle formula for cosine is Cos(x/2) = ________.
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What is the formula for converting a sum to a product in trigonometry?
What is the formula for converting a sum to a product in trigonometry?
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What types of problems can be solved using trigonometric equations?
What types of problems can be solved using trigonometric equations?
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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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