Trigonometric Formulas and Their Applications

Professor Dave introduces key trigonometric formulas, including sum and difference, double-angle, power-reducing, half-angle, product-to-sum, and sum-to-product formulas. He explains their applications and provides examples to illustrate their use, emphasizing the importance of exact values over approximations.

What is the purpose of the sum and difference formulas for sine and cosine?

• To find the sine and cosine of the sum or difference of two angles

• To find the sine and cosine of a single angle

• To approximate angles using a calculator

• To derive the unit circle

How can the cosine of five twelfths pi be calculated using known angles?

• By using a calculator

• By expressing it as a sum of quarter pi and a sixth pi

• By approximating it to the nearest whole number

• By using the Pythagorean theorem

What is the double-angle formula for cosine?

• cos(2θ) = 1 - sin²θ

• cos(2θ) = 2cosθsinθ

• cos(2θ) = cos²θ - sin²θ

• cos(2θ) = 2sinθcosθ

Which formula is used to find the sine of twice an angle?

• sin(2θ) = 2cosθsinθ

• sin(2θ) = sin²θ - cos²θ

• sin(2θ) = 1 - cos²θ

• sin(2θ) = 2sinθcosθ

What is a power-reducing formula used for?

• To calculate the hypotenuse of a triangle

• To approximate angles using a calculator

• To express sine squared in terms of cosine

• To find the tangent of an angle

How can half-angle formulas be useful?

• They simplify the unit circle

• They are used to approximate angles

• They allow evaluation of trig functions for smaller angles

• They help in finding the sine of larger angles

What is the sine of one twelfth pi expressed in terms of known angles?

• It is equal to the tangent of a sixth pi

• It is equal to the cosine of a sixth pi

• It is equal to the square root of one minus the cosine of a sixth pi over two

• It is equal to the sine of a sixth pi

What do product-to-sum formulas do?

• Find the sine of a single angle

• Approximate angles using a calculator

• Express a sum of two trig functions as a product

• Express a product of two trig functions as a sum or difference

How can sine nine X plus sine five X be expressed using sum-to-product formulas?

• As sine four X

• As two sine seven X cosine two X

• As sine fourteen X

• As cosine fourteen X

Why might it be unnecessary to memorize product-to-sum and sum-to-product formulas?

• They are only used in advanced calculus

• They are tricky to remember and not often required

• They are too complex

• They are rarely used