Problem 32 - 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 verified using the addition formula.

Step 1: Identify the Addition Formula for Cosine

The addition formula for cosine is given by: \[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]. In this exercise, we need to find \( a = x \) and \( b = \frac{\pi}{2} \).

Step 2: Substitute Values into the Addition Formula

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: Evaluate Trigonometric Values at \( \frac{\pi}{2} \)

Recall that \( \cos \frac{\pi}{2} = 0 \) and \( \sin \frac{\pi}{2} = 1 \). Substitute these values into the equation: \[ \cos(x + \frac{\pi}{2}) = \cos x \cdot 0 - \sin x \cdot 1 \].

Step 4: Simplify the Equation

Simplify the expression using the values from Step 3: \[ \cos(x + \frac{\pi}{2}) = 0 - \sin x = -\sin x \].

Step 5: Conclusion

Thus, the identity is verified and the formula \( \cos(x + \frac{\pi}{2}) = -\sin x \) is derived.

Key Concepts

Trigonometric Addition Formulas

Trigonometric addition formulas are powerful tools in trigonometry. They allow us to find the sine, cosine, or tangent of the sum of two angles.
These formulas can simplify complex trigonometric expressions and find exact values. Let's focus on the cosine addition formula.

• The cosine addition formula is: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).
• This formula provides a method to rewrite the cosine of the sum of two angles in terms of the cosine and sine of each angle separately.

When we use this formula, we break down an angle's cosine into more manageable parts. This is often helpful in both practical applications and theoretical calculations.
By getting to know these addition formulas, you'll be able to tackle more advanced problems with greater ease and confidence in trigonometry.

Understanding the Cosine Function

The cosine function, often written as \( \cos \theta \), is a fundamental element of trigonometry. It describes the cosine of an angle, which is a ratio derived from a right-angled triangle.
Cosine is associated with the x-coordinate on the unit circle. Here's how it works:

• In a right-angled triangle, \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \).
• On the unit circle, \( \cos(\theta) \) corresponds to the horizontal distance from the origin to the circle.
• Values of cosine range from \(-1\) to \(1\), representing the possible horizontal positions on the circle.

Especially important is how the cosine behaves at specific angles. For example, \( \cos(0) = 1 \) and \( \cos(\frac{\pi}{2}) = 0 \).
These values are frequently used in solving trigonometric problems and are key components in addition formulas. Understanding how the cosine function operates is crucial to comprehending trigonometry as a whole.

Exploring the Sine Function

The sine function is another core concept in trigonometry. Similar to cosine, it relates to the angles and sides of right-angled triangles and the unit circle.
The sine function, denoted as \( \sin \theta \), holds several key characteristics:

• In a right-angled triangle, \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).
• On the unit circle, \( \sin(\theta) \) corresponds to the y-coordinate, or the vertical distance from the origin.
• Sine values range between \(-1\) and \(1\).

For specific angles, such as \( \sin(0) = 0 \) and \( \sin(\frac{\pi}{2}) = 1 \), the sine function provides values crucial for solving equations and identities involving angles.
These known values are essential when using trigonometric identities like the addition formula for cosine, where \( \sin(\frac{\pi}{2}) \) plays a pivotal role in simplification of expressions.