Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-36: Verify the reduction formula. $$ \sin \left(x+\frac{\pi}{2}\right)=\cos x $$

Step-by-Step Solution

Answer

The identity \( \sin \left(x + \frac{\pi}{2}\right) = \cos x \) is verified using trigonometric values.

Step 1: Understand the Formula

We need to verify the trigonometric identity \( \sin \left(x + \frac{\pi}{2}\right) = \cos x \). This implies demonstrating that when you shift the angle \( x \) by \( \frac{\pi}{2} \), the sine function behaves like the cosine function.

Step 2: Recall the Identity of Sine and Cosine

Recall the identity for the sine of a sum of two angles: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). We'll use this identity with \( a = x \) and \( b = \frac{\pi}{2} \).

Step 3: Apply the Identity

Substitute into the formula: \( \sin \left(x + \frac{\pi}{2}\right) = \sin x \cos \frac{\pi}{2} + \cos x \sin \frac{\pi}{2} \).

Step 4: Evaluate Trigonometric Values

Calculate the trigonometric values: \( \cos \frac{\pi}{2} = 0 \) and \( \sin \frac{\pi}{2} = 1 \). Substitute these into the equation: \( \sin \left(x + \frac{\pi}{2}\right) = \sin x \cdot 0 + \cos x \cdot 1 \).

Step 5: Simplify the Expression

Simplify the expression: \( \sin \left(x + \frac{\pi}{2}\right) = 0 + \cos x = \cos x \). Thus, the identity \( \sin \left(x + \frac{\pi}{2}\right) = \cos x \) is verified.

Key Concepts

Sine Function

The sine function is a basic trigonometric function that describes the relationship between an angle and its position on the unit circle. Consider a right triangle inscribed in a circle with hypotenuse as the radius. The sine of an angle \(x\) in this triangle is the ratio of the length of the opposite side to the hypotenuse. In the context of a unit circle, the hypotenuse is always 1. Consequently, sine is equal to the length of the vertical line from the point on the circle to the \(x\)-axis.Key properties of the sine function include:

• Its domain is all real numbers, captured as \([-\infty, \infty]\).
• It has a range between -1 and 1, since it represents a ratio of side lengths.
• The sine function is periodic, repeating every \(2\pi\).

When we add \(\frac{\pi}{2}\) to the angle in the sine function, we're effectively rotating 90 degrees clockwise on the unit circle. This change directly transforms the behavior of the sine function into that of the cosine, demonstrating one of the key properties of trigonometric functions.

Cosine Function

The cosine function, much like the sine, connects angles with coordinates on the unit circle. In a right triangle, the cosine of an angle \(x\) is the ratio of the adjacent side to the hypotenuse. Again, on a unit circle, this becomes the horizontal distance from the origin to the point on the circle. Some essential characteristics of the cosine function include:

• Its domain includes all real numbers.
• Cosine falls within the same range as sine, between -1 and 1.
• The function also exhibits periodicity with a period of \(2\pi\).

Unlike sine, the cosine function starts at its maximum value of 1 when \(x = 0\). Thus, it is regarded as a phase-shifted version of sine. Shifting the sine by \(\frac{\pi}{2}\) illustrates this direct correspondence because this operation moves the sine graph to align it with the cosine's start point.

Angle Addition Formulas

Angle addition formulas are vital in understanding how trigonometric functions respond to the addition of angles. These formulas express trigonometric functions as products and sums of functions of individual angles. The sine addition formula, \( \sin(a + b) = \sin a \cos b + \cos a \sin b \), is particularly helpful in the validation of identities and transformations.How it works:

• The first term, \(\sin a \cos b\), captures the influence of both angles on the vertical component.
• The second term, \(\cos a \sin b\), represents their effect on the horizontal component.

In the exercise, by substituting \(a = x\) and \(b = \frac{\pi}{2}\), this addition formula demonstrates how the sine of angle \(x + \frac{\pi}{2}\) shifts to become the cosine of \(x\). Understanding these transformations through angle addition is crucial for mastering trigonometric identities.