Problem 15 - Precalculus Essentials

Question

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working \(5-8\) in the Concept and Vocabulary Check. Express each sum or difference as a product. If possible, find this product's exact value. $$\sin x+\sin 2 x$$

Step-by-Step Solution

Answer

The sum \(\sin(x) + \sin(2x)\) can be expressed as a product: \(2\sin( \frac{3x}{2}) \cos(\frac{x}{2})\). An exact value cannot be determined without specific values for \(x\).

Step 1: Identify the Relevant Trigonometric Identity

The trigonometric identity to be used here is: \(\sin (a) + \sin (b) = 2\sin \frac{1}{2}(a+b) \cos \frac{1}{2}(a-b)\). In there, \(a\) and \(b\) are the angles, which in this case are \(x\) and \(2x\), respectively.

Step 2: Apply the Trigonometric Identity to the Problem

Substitute \(x\) and \(2x\) into the formula obtained in step 1: \(2\sin \frac{1}{2}(x+2x) \cos \frac{1}{2}(x-2x) = 2\sin( \frac{3x}{2}) \cos( -\frac{x}{2}) \). It should be noted that \(\cos\) of an angle is equal to the \(\cos\) of its negative. Therefore, the formula becomes \(2\sin( \frac{3x}{2}) \cos(\frac{x}{2})\).

Step 3: Determine the Exact Value

In this case, it is not possible to find the exact value of the formula because it depends on the variable \(x\). Without specific values for \(x\), the exact value cannot be determined. However, we have successfully expressed the original sum as a product.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is essential for simplifying expressions and solving trigonometric equations. These identities are equations that hold true for all values of the variables involved. They can relate the trigonometric functions—sine, cosine, tangent, cotangent, secant, and cosecant—to one another.
One fundamental set of identities are the Pythagorean identities, stating that for any angle \theta, \( \sin^2\theta + \cos^2\theta = 1 \) and the variants obtained by dividing through by \( \sin^2\theta \) or \( \cos^2\theta \) respectively.
Another key set of identities are the angle sum and difference identities, which allow us to express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. We've used one of these identities in the exercise to transform a sum into a product.

Sine Function

The sine function is one of the primary trigonometric functions and is abbreviated as \( \sin \). It is defined for any angle in a right-angled triangle as the ratio of the length of the opposite side to the length of the hypotenuse. Additionally, it can be represented as a point on the unit circle corresponding to an angle, describing the y-coordinate.

• <\( \sin 0^\circ = 0 \) is a unique property reflecting the origin point on the unit circle.
• <\( \sin 90^\circ = 1 \) represents the maximum value of the function on the y-axis.
• The sine function is periodic with a period of <\( 2\pi \) radians or 360 degrees, meaning the values repeat after this interval.

By using the sine function in conjunction with trigonometric identities, we can simplify complex trigonometric expressions, like in our original problem where the sum of \( \sin x \) and \( \sin 2x \) is converted to a product.

Cosine Function

Complementary to the sine function, the \( \cos \) (cosine) function plays a crucial role in trigonometry. It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle, or the x-coordinate of a point on the unit circle.

• <\( \cos 0^\circ = 1 \) signifies the starting point on the unit circle's x-axis.
• <\( \cos 90^\circ = 0 \) because at 90 degrees, the point on the unit circle moves entirely onto the y-axis.
• The cosine function also has a period of <\( 2\pi \) radians (360 degrees) like the sine function.

Its relationship with sine through the Pythagorean identity <\( \sin^2\theta + \cos^2\theta = 1 \) is quintessential in solving trigonometric problems. Notably, as seen in the exercise, the cosine function is even, meaning <\( \cos\theta = \cos(-\theta) \), which was utilized to simplify the expression.

Angle Addition Formulas

Angle addition formulas are pivotal when dealing with the trigonometric functions of sums and differences of angles. They state that:

• <\( \sin(a + b) = \sin a \cos b + \cos a \sin b \) for the sine of the sum of two angles.
• <\( \cos(a + b) = \cos a \cos b - \sin a \sin b \) for the cosine of the sum.
• The corresponding formulas for differences of angles are obtained by changing the sign of the second angle.

In our exercise, we employ a variant of these formulas: <\( \sin a + \sin b = 2\sin \frac{1}{2}(a+b) \cos \frac{1}{2}(a-b) \) to convert the sum of two sines into a product form. This transformation is a valuable skill as it simplifies the original trigonometric expression and prepares it for further manipulation or solving, particularly when integrating trigonometric functions or finding their values.