Problem 15 - Precalculus Essentials

Question

In Exercises \(15-22,\) write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. $$2 \sin 15^{\circ} \cos 15^{\circ}$$

Step-by-Step Solution

Answer

The expression \(2 \sin 15^{\circ} \cos 15^{\circ}\) equals \(0.5\).

Step 1 - Use the Double angle formula for sine

The first step is to use the double angle identity for sine. In this case, we transform \(2 \sin x \cos x = \sin 2x\). Therefore, \(2 \sin 15^{\circ}\cos 15^{\circ}\) turns into \(\sin 30^{\circ}\).

Step 2 - Evaluate the expression

The next step is evaluating the new expression. The sine of 30 degrees, \(\sin 30^{\circ}\), is equal to \(0.5\) in the unit circle.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is fundamental in solving many mathematical problems that involve angle measures. Identities are equations that hold true for all values of the variable involved. In trigonometry, these identities often involve the functions sine, cosine, and tangent, and they are used to simplify expressions, solve trigonometric equations, and prove other mathematical properties.
Double angle identities are a type of trigonometric identity that relate the sine, cosine, or tangent of double angles to the trigonometric functions of a single angle. As seen in the exercise, the identity for the sine of a double angle is particularly useful:

• \(2 \sin x \cos x = \sin 2x\)

This powerful identity allows the simplification of trigonometric expressions, making it easier to find exact values and solve equations.

Sine Function

The sine function is one of the primary trigonometric functions and is abbreviated as \(\sin\). It relates the angle of a right-angled triangle to the ratio of the length of the opposite side to the hypotenuse. In a unit circle context, where the hypotenuse is always 1, the sine of an angle is the y-coordinate of the point where the terminal side of the angle intersects the circle.
In the exercise provided, the sine function is used to simplify an expression involving a double angle. By recognizing patterns and applying identities, you can often turn a complex problem into a simpler one. For example:

Conversion to Double Angle

By using the double angle identity, \(2 \sin 15^\circ \cos 15^\circ\) becomes \(\sin 30^\circ\). This showcases the convenience of the sine function in conjunction with identities to make calculations more manageable.

Cosine Function

The cosine function, symbolized as \(\cos\), is another core trigonometric function. It is closely related to the sine function but instead measures the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. In the framework of the unit circle, the cosine of an angle gives us the x-coordinate of the intersection point on the circle's circumference.
Just like the sine function, the cosine function has associated identities that simplify expressions. For example, the double angle formula for cosine, which is not used in the given exercise, can be expressed in multiple forms:

• \(\cos 2x = \cos^2 x - \sin^2 x\)
• \(\cos 2x = 2\cos^2 x - 1\)
• \(\cos 2x = 1 - 2\sin^2 x\)

These forms can greatly aid in various problem-solving scenarios much like we saw with the sine double angle identity.

Trigonometric Formulas

Trigonometric formulas are the engines that drive much of trigonometry's problem-solving capacity. They encompass a variety of identities and equations, including those for angle sum and difference, double and half angles, and product-to-sum formulas.
Through consistent practice and application of these formulas, complicated trigonometric problems can often be reduced to something much simpler. For example, the double angle formulas, a subset of trigonometric formulas, were put to use in the given exercise, transforming a product of sine and cosine into a single function of a double angle.

Advantages of Trigonometric Formulas

Using the right formula at the appropriate time can save effort and lead to more accurate results. It also helps improve a student's mathematical intuition and problem-solving skills. As seen with the original problem, correctly applying the double angle identity for sine made it possible to find the exact value swiftly, showcasing the power and utility of trigonometric formulas in mathematics.