Problem 26 - Precalculus Mathematics for Calculus

Question

Prove the identity. $$\cos \left(x-\frac{\pi}{2}\right)=\sin x$$

Step-by-Step Solution

Answer

The identity is proved: \(\cos(x - \frac{\pi}{2}) = \sin(x)\).

Step 1: Recall the Cosine and Sine Difference Formulas

The cosine of a difference \( \cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b) \). The exercise states:\( \cos\left( x - \frac{\pi}{2} \right) \), where \(a = x\) and \(b = \frac{\pi}{2}\). Use this formula to expand the expression.

Step 2: Expand Using the Cosine Difference Formula

Expand \( \cos\left( x - \frac{\pi}{2} \right) \) using the cosine difference formula:\[\cos(x - \frac{\pi}{2}) = \cos(x)\cos\left(\frac{\pi}{2}\right) + \sin(x)\sin\left(\frac{\pi}{2}\right)\]

Step 3: Substitute the Known Trigonometric Values

We know the values:\( \cos\left(\frac{\pi}{2}\right) = 0 \) and \( \sin\left(\frac{\pi}{2}\right) = 1 \).Substitute these into the equation:\[\cos(x - \frac{\pi}{2}) = \cos(x) \times 0 + \sin(x) \times 1\]

Step 4: Simplify the Expression

Since \( \cos(x) \times 0 = 0 \), the expression simplifies to: \[\cos(x - \frac{\pi}{2}) = \sin(x)\] Thus, proving the identity.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It is defined based on the angle and the adjacent side in a right-angled triangle. The function is represented as \(\cos(\theta)\), where \(\theta\) is the angle. In the unit circle, the cosine of an angle corresponds to the x-coordinate of a point on the circle.

Some properties of the cosine function include:

• It has a range from -1 to 1.
• Its period is \(2\pi\), meaning the function repeats every \(2\pi\) radians.
• It is an even function, which implies that \(\cos(-\theta) = \cos(\theta)\).
• Maximum value is 1, occurring at \(\theta = 0, 2\pi, 4\pi, \ldots\)

The cosine function plays a crucial role in trigonometry and is used to model periodic phenomena, such as waves and oscillations.

Understanding how cosine interacts with other trigonometric functions is vital for solving trigonometric identities and equations.

Sine Function

The sine function is another basic trigonometric function. It focuses on the angle and the opposite side in a right-angled triangle, represented as \(\sin(\theta)\). When we look at the unit circle, the sine of an angle is the y-coordinate of the point on the circle.

Key features of the sine function include:

• Also has a range from -1 to 1.
• Its period is \(2\pi\).
• It is an odd function, indicating \(\sin(-\theta) = -\sin(\theta)\).
• Highest value is 1, occurring at \(\theta = \frac{\pi}{2}, \frac{5\pi}{2}, \ldots\)

The sine function is essential for exploring relationships in waveforms and oscillations, similar to cosine.

It's important to frequently compare the sine and cosine functions, as their differences and similarities are often exploited in solving problems involving trigonometric identities and calculations.

Angle Subtraction Formula

The angle subtraction formula is a powerful tool for simplifying trigonometric expressions and solving equations. For the cosine function, it is expressed as \(\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)\). This formula allows us to express the cosine of a difference between two angles in terms of the cosines and sines of these angles separately.

Why is this formula useful?

• It helps transform complex trigonometric equations into simpler, more manageable forms.
• It is essential for proving trigonometric identities and facilitating integrations and derivations.

In our exercise, the formula was used by setting \(a = x\) and \(b = \frac{\pi}{2}\). By substituting these values, and knowing the specific trigonometric values, the expression simplified readily to \(\sin(x)\), showcasing the utility of trigonometric identities.

Mastering the angle subtraction formula is vital for tackling more advanced problems in trigonometry and calculus.