Problem 26 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Using trigonometric identities, we find that \( \cos(x - \frac{\pi}{2}) = \sin x \).

Step 1: Understand the identity to prove

The identity to prove is \( \cos \left( x - \frac{\pi}{2} \right) = \sin x \). This means showing that if you subtract \( \frac{\pi}{2} \) from an angle \( x \), the cosine of that result equals the sine of the original angle \( x \).

Step 2: Recall the cosine subtraction formula

The formula for the cosine of a difference is \( \cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b) \). In this problem, we set \( a = x \) and \( b = \frac{\pi}{2} \).

Step 3: Apply the formula to the identity

Substituting \(a\) and \(b\) into the difference formula gives us: \[ \cos \left( x - \frac{\pi}{2} \right) = \cos x \cos \frac{\pi}{2} + \sin x \sin \frac{\pi}{2} \]

Step 4: Simplify using values of trigonometric functions

We know \( \cos \frac{\pi}{2} = 0 \) and \( \sin \frac{\pi}{2} = 1 \). Substituting these values in, the expression simplifies to: \[ \cos x \cdot 0 + \sin x \cdot 1 = \sin x \]

Step 5: Verify and conclude

After substitution and simplification, we obtain \( \sin x \), which matches the right side of our original identity \( \sin x \). This confirms that the identity \( \cos \left( x - \frac{\pi}{2} \right) = \sin x \) is correct.

Key Concepts

Cosine Subtraction Formula

The cosine subtraction formula is a key tool in trigonometry. It helps us find the cosine of one angle minus another. This formula is especially useful when you need to simplify expressions or prove identities.

The formula is written as:

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

In our specific problem, we are asked to prove that \( \cos \left( x - \frac{\pi}{2} \right) = \sin x \). By setting \( a = x \) and \( b = \frac{\pi}{2} \), we can substitute them into the formula. This gives us:

• \( \cos(x)\cos\left(\frac{\pi}{2}\right) + \sin(x)\sin\left(\frac{\pi}{2}\right) \)

Here, understanding the values of \( \cos\left(\frac{\pi}{2}\right) \) and \( \sin\left(\frac{\pi}{2}\right) \) is crucial to simplifying the problem correctly.

Sine Function

The sine function is one of the primary trigonometric functions. It describes the y-coordinate on the unit circle corresponding to a given angle. This function has a period of \( 2\pi \), meaning it repeats itself every \( 2\pi \) radians.

Sine values range from \(-1\) to \(1\) as the angle travels around the unit circle.

• \( \sin\left(0\right) = 0 \)
• \( \sin\left(\frac{\pi}{2}\right) = 1 \)
• \( \sin\left(\pi\right) = 0 \)
• \( \sin\left(\frac{3\pi}{2}\right) = -1 \)
• \( \sin\left(2\pi\right) = 0 \)

In our identity, the sine function is utilized because when you compute \( \cos\left(x - \frac{\pi}{2}\right) \), you eventually arrive at \( \sin(x) \) due to the properties of the sine and cosine functions.

Trigonometric Functions

Trigonometric functions include sine, cosine, tangent, and their reciprocals. They are based on the properties of triangles, especially right triangles, and the unit circle.

Understanding these functions can help in solving various mathematical problems:

• \( \sin \) (sine) - the opposite side over the hypotenuse
• \( \cos \) (cosine) - the adjacent side over the hypotenuse
• \( \tan \) (tangent) - the opposite side over the adjacent side

These functions have specific identities and formulas that help in their manipulation and simplification. For example, one important identity is \( \sin^2(x) + \cos^2(x) = 1 \), which explains the relationship between sine and cosine.

Moreover, these functions' values at certain key angles (like \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \), and \( 2\pi \)) provide insights into equations and identities. In the identity \( \cos \left( x - \frac{\pi}{2} \right) = \sin x \), the interplay between sine and cosine and their distinctive patterns makes this identity hold true.