Problem 34 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

\(\sin(x-\frac{\pi}{2}) = -\cos x\) is derived by using the sine difference formula and evaluating trigonometric values.

Step 1: Identify the Addition Formula

The problem involves the expression \( \sin(x - \frac{\pi}{2}) \). We need to use the sine of a difference addition formula: \( \sin(a-b) = \sin a \cos b - \cos a \sin b \).

Step 2: Plug in Values

For \( a = x \) and \( b = \frac{\pi}{2} \), substitute these into the sine of a difference formula: \( \sin(x-\frac{\pi}{2}) = \sin x \cos\frac{\pi}{2} - \cos x \sin\frac{\pi}{2} \).

Step 3: Evaluate Trigonometric Functions

Evaluate the trigonometric functions for \( \frac{\pi}{2} \): \( \cos\frac{\pi}{2} = 0 \) and \( \sin\frac{\pi}{2} = 1 \). Substitute these back into the expression: \( \sin(x-\frac{\pi}{2}) = \sin x \times 0 - \cos x \times 1 \).

Step 4: Simplify the Expression

Now simplify the expression: \( \sin(x-\frac{\pi}{2}) = 0 - \cos x = -\cos x \).

Step 5: Conclusion

We derived the identity \( \sin(x-\frac{\pi}{2}) = -\cos x \) using addition formulas.

Key Concepts

Addition Formulas

The addition formulas are key tools in trigonometry that help in breaking down complex trigonometric expressions into simpler components.
For the sine function, the addition formula states:

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

These formulas allow us to express the sine of the sum or difference of two angles in terms of the sines and cosines of the individual angles.
They are particularly useful in deriving other identities, as we will see with angle subtraction identities.
This step ensures that we can manipulate trigonometric expressions systematically.

Sine Function

The sine function is an essential trigonometric function often abbreviated as \( \sin \).
It is the y-coordinate of the point on the unit circle corresponding to a given angle.
The sine function has a range from -1 to 1 and is periodic with a period of \( 2\pi \).

• Key Feature: At \( \frac{\pi}{2} \), the sine function equals 1, which is crucial in calculations such as our example identity.

This function can describe oscillations such as sound waves and light waves and is fundamental to understanding phenomena in both engineering and physics.

Cosine Function

The cosine function, expressed as \( \cos \), is another fundamental trigonometric function.
It represents the x-coordinate of a point on the unit circle for a particular angle.
Like sine, the cosine function varies between -1 and 1 and has a period of \( 2\pi \).

• Important Value: The cosine of \( \frac{\pi}{2} \) is 0, which directly impacts calculations involving the addition or subtraction of angles.

Understanding how cosine behaves in different quadrants helps predict the sign and value of the cosine function, which is pivotal when simplifying expressions.

Angle Subtraction Identities

Angle subtraction identities follow the pattern of addition formulas, allowing us to calculate the trigonometric functions of differences between angles.
The angle subtraction identity for sine, as used in our solution, is:

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

This identity was crucial in deriving the expression \( \sin(x-\frac{\pi}{2}) = -\cos x \).
By plugging in the correct values (e.g., \( a = x \), \( b = \frac{\pi}{2} \)), we decomposed the expression into known values and simplified them.
Such identities are useful in various applications, from calculating angles in design to solving physics problems.