Problem 62 - Precalculus with Limits

Question

In Exercises 61 - 70, prove the identity. \( \sin\left(\dfrac{\pi}{2} + x\right) = \cos x \)

Step-by-Step Solution

Answer

Yes, the given equation \( \sin\left(\dfrac{\pi}{2} + x\right) = \cos x \) is indeed a valid trigonometric identity.

Step 1: Identify the Sine Addition Formula

The formula for sine of the sum of two angles is:\[\sin(a + b) = \sin a \cos b + \cos a \sin b\]This will be applied to the given expression to simplify it.

Step 2: Apply the Formula and Simplify

Apply the sine addition formula to the given expression, substituting \(a = \pi/2\) and \(b = x\). We get: \[\sin\left(\dfrac{\pi}{2} + x\right) = \sin \dfrac{\pi}{2} \cos x + \cos \dfrac{\pi}{2} \sin x = \cos x \]The last step of simplification comes from knowing that \(\sin \dfrac{\pi}{2} = 1\) and \(\cos \dfrac{\pi}{2} = 0\). So the second term \(\cos \dfrac{\pi}{2} \sin x\) cancels out, and the remaining term gives us exactly what we started with, \(\cos x\).

Step 3: Conclude

The computation shows that the two expressions \( \sin\left(\dfrac{\pi}{2}+x\right) \) and \( \cos x \) are indeed equal. Therefore, the given equation represents a valid trigonometric identity.

Key Concepts

Sine Addition Formula

The sine addition formula is a fundamental tool in trigonometry, especially when working with angle sums. It states that for any two angles, \(a\) and \(b\), the sine of their sum is:

• \(\sin(a + b) = \sin a \cos b + \cos a \sin b\)

Using this formula allows you to simplify complex trigonometric expressions by breaking them down into sine and cosine components of the individual angles. In the problem, this concept was used to transform \(\sin\left(\frac{\pi}{2} + x\right)\) into something more manageable. By recognizing \(a\) as \(\frac{\pi}{2}\) and \(b\) as \(x\), the identity simplifies using known values for sine and cosine at specific angles. This formula is key in proving trigonometric identities and understanding how angles interact in trigonometric functions.

Angle Sum Identities

Angle sum identities are a powerful set of formulas used in trigonometry for expressing the trigonometric functions of sums or differences of angles. Besides the sine addition formula, they also include:

• Cosine addition and subtraction: \(\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b\)
• Tangent addition and subtraction: \(\tan(a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b}\)

These identities help simplify expressions involving compound angles. In the particular exercise, the sine addition identity allowed simplifying the expression \(\sin\left(\frac{\pi}{2} + x\right)\) directly into a basic trigonometric function, \(\cos x\), by simplifying terms using known trigonometric values of special angles.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are foundational elements in mathematics, particularly in circles and angles. They enable the analysis of periodic phenomena and are pivotal in geometry and physics. These functions include:

• Sine: \(\sin(x)\) - the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine: \(\cos(x)\) - the ratio of the adjacent side to the hypotenuse.
• Tangent: \(\tan(x)\) - the ratio of the sine to the cosine, or opposite to adjacent.

In the given problem, recognizing the values of \(\sin\left(\frac{\pi}{2}\right)=1\) and \(\cos\left(\frac{\pi}{2}\right)=0\) was crucial in simplifying the sine addition formula application. Mastery of trigonometric functions and understanding their properties and relationships is vital for solving trigonometric identities and proving complex mathematical statements.