Trigonometric Formulas and Equations

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

What is the double angle formula for sine?

Sin(2x) = 2Sin(x)Cos(x) Signup and view all the answers

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

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

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

Cos(a + b) = Cos(a)Cos(b) - Sin(a)Sin(b) Signup and view all the answers

What types of problems can be solved using trigonometric equations?

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

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.

Description

Test your understanding of trigonometric identities, Pythagorean concepts, and the characteristics of even and odd trig functions. This quiz covers essential formulas including sum and difference, double and half angle formulas, as well as solving various trigonometric equations.