Solving Trigonometry Problems with Product-to-Sum Formulas

Let's evaluate the given expressions for x and y and then find their product xy. We are given:

x = sin 70° ⋅ sin 50°

y = cos 60° ⋅ cos 80°

We will use the product-to-sum trigonometric identities to simplify the expressions for x and y.

Using Product-to-Sum Formulas for x

The product-to-sum formula for two sine functions is:

\(2 \sin A \sin B = \cos(A - B) - \cos(A + B)\)

For x = sin 70° ⋅ sin 50°, let A = 70° and B = 50°. So, \(2x = 2 \sin 70° \sin 50°\).

Applying the formula:

\(2x = \cos(70° - 50°) - \cos(70° + 50°)\)

\(2x = \cos 20° - \cos 120°\)

We know that \(\cos 120° = \cos(180° - 60°) = -\cos 60°\). The value of \(\cos 60°\) is \(1/2\).

So, \(\cos 120° = -1/2\).

Substituting this value back into the equation for \(2x\):

\(2x = \cos 20° - (-1/2)\)

\(2x = \cos 20° + 1/2\)

Now, divide by 2 to find x:

\(x = \frac{1}{2} \cos 20° + \frac{1}{4}\)

Evaluating y

The expression for y is given as:

\(y = \cos 60° \sdot \cos 80°\)

We know the exact value of \(\cos 60°\) is \(1/2\).

Substitute this value into the expression for y:

\(y = \frac{1}{2} \cos 80°\)

Calculating the Product xy

Now we need to find the product xy by multiplying the expressions we found for x and y:

\(xy = \left(\frac{1}{2} \cos 20° + \frac{1}{4}\right) \left(\frac{1}{2} \cos 80°\right)\)

Distribute the term \(\left(\frac{1}{2} \cos 80°\right)\) inside the first parenthesis:

\(xy = \left(\frac{1}{2} \cos 20°\right) \left(\frac{1}{2} \cos 80°\right) + \left(\frac{1}{4}\right) \left(\frac{1}{2} \cos 80°\right)\)

\(xy = \frac{1}{4} \cos 20° \cos 80° + \frac{1}{8} \cos 80°\)

We have a product of cosines term, \(\cos 20° \cos 80°\). Let's use another product-to-sum formula:

\(2 \cos A \cos B = \cos(A + B) + \cos(A - B)\)

For \(\cos 20° \cos 80°\), let A = 80° and B = 20° (order doesn't matter for the sum, but for difference, we can take the larger angle first to keep it positive, or remember \(\cos(-\theta) = \cos \theta\)). Let's use A=80°, B=20°:

\(2 \cos 80° \cos 20° = \cos(80° + 20°) + \cos(80° - 20°)\)

\(2 \cos 80° \cos 20° = \cos 100° + \cos 60°\)

We know \(\cos 60° = 1/2\). Also, \(\cos 100° = \cos(180° - 80°) = -\cos 80°\).

So, \(2 \cos 80° \cos 20° = -\cos 80° + 1/2\)

Divide by 2:

\(\cos 80° \cos 20° = \frac{1}{2} \left(-\cos 80° + \frac{1}{2}\right) = -\frac{1}{2} \cos 80° + \frac{1}{4}\)

Now substitute this back into the expression for xy:

\(xy = \frac{1}{4} \left(-\frac{1}{2} \cos 80° + \frac{1}{4}\right) + \frac{1}{8} \cos 80°\)

Distribute the \(1/4\):

\(xy = -\frac{1}{8} \cos 80° + \frac{1}{16} + \frac{1}{8} \cos 80°\)

The terms \(-\frac{1}{8} \cos 80°\) and \(+\frac{1}{8} \cos 80°\) cancel each other out.

\(xy = \frac{1}{16}\)

Thus, the value of xy is \(1/16\).

Summary of Steps:

• Defined x and y using the given trigonometric expressions.
• Used the product-to-sum formula \(2 \sin A \sin B = \cos(A - B) - \cos(A + B)\) to simplify x.
• Evaluated \(\cos 120°\) and found x in terms of \(\cos 20°\).
• Evaluated y using the known value of \(\cos 60°\).
• Multiplied the expressions for x and y.
• Used the product-to-sum formula \(2 \cos A \cos B = \cos(A + B) + \cos(A - B)\) for the resulting product term \(\cos 20° \cos 80°\).
• Evaluated \(\cos 100°\) and \(\cos 60°\) in the simplified product term.
• Substituted back and simplified the expression for xy, cancelling terms.

Revision Table: Key Trigonometry Concepts

Additional Information on Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables for which both sides of the equation are defined. They are crucial for simplifying expressions, solving equations, and evaluating exact values.

Common types of identities include:

• Reciprocal Identities (e.g., \(\csc \theta = 1/\sin \theta\))
• Quotient Identities (e.g., \(\tan \theta = \sin \theta / \cos \theta\))
• Pythagorean Identities (e.g., \(\sin^2 \theta + \cos^2 \theta = 1\))
• Sum and Difference Identities (e.g., \(\sin(A+B) = \sin A \cos B + \cos A \sin B\))
• Double and Half Angle Identities
• Product-to-Sum and Sum-to-Product Identities (used in this problem)

Mastering these identities is key to success in trigonometry. The product-to-sum identities are particularly useful when dealing with products of sine and cosine functions and can often simplify expressions involving specific angles.