Problem 31 - Thomas Calculus

Question

Use the addition formulas to derive the identities in Exercises \(31-36\) $$\cos \left(x-\frac{\pi}{2}\right)=\sin x$$

Step-by-Step Solution

Answer

The identity \( \cos(x - \frac{\pi}{2}) = \sin x \) is derived using the cosine addition formula.

Step 1: Understanding the Addition Formula for Cosine

The addition formula for cosine states that \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). We need to use this formula to express \( \cos(x - \frac{\pi}{2}) \).

Step 2: Applying the Addition Formula to Our Problem

Substitute \( a = x \) and \( b = -\frac{\pi}{2} \) into the addition formula: \( \cos(x - \frac{\pi}{2}) = \cos x \cos(-\frac{\pi}{2}) - \sin x \sin(-\frac{\pi}{2}) \).

Step 3: Evaluating the Trigonometric Functions at Specific Angles

Recall the trigonometric values: \( \cos(-\frac{\pi}{2}) = 0 \) and \( \sin(-\frac{\pi}{2}) = -1 \). Substitute these into the expression: \( \cos(x - \frac{\pi}{2}) = \cos x \cdot 0 - \sin x \cdot (-1) \).

Step 4: Simplifying the Expression

Simplify the expression to get \( \cos(x - \frac{\pi}{2}) = 0 - (-\sin x) = \sin x \). This shows that the expression \( \cos(x - \frac{\pi}{2}) \) simplifies to \( \sin x \).

Step 5: Conclusion

The derived identity \( \cos(x - \frac{\pi}{2}) = \sin x \) has been confirmed by using the addition formula and evaluating trigonometric functions.

Key Concepts

Addition Formula

The addition formula for trigonometric functions is a key tool in trigonometry. It allows us to express trigonometric functions of sums or differences of angles in terms of the trigonometric functions of the individual angles. For the cosine function, the addition formula is given by:

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

This formula is extremely useful when dealing with complex trigonometric expressions. In the provided exercise, we specifically deal with the identity \( \cos(x - \frac{\pi}{2}) \). By substituting \( a = x \) and \( b = -\frac{\pi}{2} \), we transform this problem into evaluating a known formula, thereby simplifying our work with these functions. Understanding and applying the addition formula can aid greatly in deriving and proving trigonometric identities.

Trigonometric Functions

Trigonometric functions are the backbone of trigonometry and are crucial in various fields like physics, engineering, and mathematics. These functions, like sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. Important to this exercise are:

• Cosine: \( \cos \theta \)
• Sine: \( \sin \theta \)

The values of these functions at specific angles like \( 0, \frac{\pi}{2}, \pi, \text{and} -\frac{\pi}{2} \) are fundamental. For example:

• \( \cos(-\frac{\pi}{2}) = 0 \)
• \( \sin(-\frac{\pi}{2}) = -1 \)

In our exercise, these values were directly utilized to simplify the expression using the addition formula for cosine. Mastering these fundamental angles and their corresponding trigonometric values is essential for proficiently solving trigonometric problems.

Angle Subtraction

The concept of angle subtraction is an integral part of understanding how to manipulate angles in trigonometric identities. Angle subtraction can be thought of as finding the result of one angle minus another. The subtraction of angles relates directly to:

• The addition formula, which can be tweaked to apply to subtraction by considering the addition of a negative angle.
• The fact that subtracting an angle involves considering its supplementary properties.

In our exercise, we dealt with \( x - \frac{\pi}{2} \), which can be interpreted as adding a negative angle to \( x \). By applying the addition formula to the difference of angles, \( \cos(x - \frac{\pi}{2}) \), we showcased how subtraction translates into addition in the context of trigonometric identities. This allows simplification and solution of problems more efficiently.