Trigonometric Identities and Formulas

The video tutorial covers product-to-sum and sum-to-product trigonometric formulas, providing explanations and examples for each. It demonstrates how to simplify trigonometric expressions and verify identities using these formulas. The tutorial emphasizes understanding the application of these formulas in solving trigonometric problems.

Which formula represents the product of sine and cosine in terms of a sum?

• sine(alpha) * sine(beta) = 1/2 [cos(alpha - beta) - cos(alpha + beta)]

• sine(alpha) * cosine(beta) = 1/2 [sine(alpha + beta) + sine(alpha - beta)]

• cosine(alpha) * cosine(beta) = 1/2 [cos(alpha - beta) + cos(alpha + beta)]

• cosine(alpha) * sine(beta) = 1/2 [sine(alpha + beta) - sine(alpha - beta)]

How would you simplify the expression sine(7x) * sine(4x) using product-to-sum formulas?

• 1/2 [cos(3x) - cos(11x)]

• 1/2 [sine(3x) + sine(11x)]

• 1/2 [cos(3x) + cos(11x)]

• 1/2 [sine(3x) - sine(11x)]

What is the simplified form of sine(9x) * cosine(3x) using product-to-sum formulas?

• 1/2 [cosine(12x) + cosine(6x)]

• 1/2 [sine(12x) - sine(6x)]

• 1/2 [cosine(12x) - cosine(6x)]

• 1/2 [sine(12x) + sine(6x)]

What is the sum-to-product formula for sine(alpha) + sine(beta)?

• 2 cosine((alpha + beta)/2) sine((alpha - beta)/2)

• 2 cosine((alpha - beta)/2) sine((alpha + beta)/2)

• 2 sine((alpha + beta)/2) cosine((alpha - beta)/2)

• 2 sine((alpha - beta)/2) cosine((alpha + beta)/2)

Which formula is used to convert cosine(alpha) - cosine(beta) into a product?

• Negative 2 sine((alpha + beta)/2) sine((alpha - beta)/2)

• 2 cosine((alpha + beta)/2) sine((alpha - beta)/2)

• 2 sine((alpha + beta)/2) cosine((alpha - beta)/2)

• Negative 2 cosine((alpha + beta)/2) sine((alpha - beta)/2)

Which formula is used to convert sine(alpha) - sine(beta) into a product?

• 2 sine((alpha - beta)/2) cosine((alpha + beta)/2)

• 2 sine((alpha + beta)/2) cosine((alpha - beta)/2)

• 2 cosine((alpha - beta)/2) sine((alpha + beta)/2)

• 2 cosine((alpha + beta)/2) sine((alpha - beta)/2)

Using sum-to-product formulas, how would you simplify sine(8x) + sine(3x)?

• 2 sine(5x/2) cosine(11x/2)

• 2 cosine(5x/2) sine(11x/2)

• 2 cosine(11x/2) sine(5x/2)

• 2 sine(11x/2) cosine(5x/2)

What is the simplified form of cosine(11x) + cosine(3x) using sum-to-product formulas?

• 2 sine(4x) sine(7x)

• 2 cosine(4x) cosine(7x)

• 2 sine(7x) sine(4x)

• 2 cosine(7x) cosine(4x)

What is the result of simplifying sine(75) + sine(15) using sum-to-product formulas?

• Square root of 5 divided by 2

• Square root of 2 divided by 2

• Square root of 6 divided by 2

• Square root of 3 divided by 2

How do you verify the identity sine(x) + sine(3x) over cosine(x) + cosine(3x) equals tangent(2x)?

• Use product-to-sum formulas for both numerator and denominator

• Use only sum-to-product formulas for the denominator

• Use only product-to-sum formulas for the numerator

• Use sum-to-product formulas for both numerator and denominator