Problem 25 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The identity \( \sin(x-\frac{\pi}{2}) = -\cos x \) is proven using the angle subtraction formula and known trigonometric values.

Step 1: Understand the Identity

The identity we need to prove is \( \sin \left(x-\frac{\pi}{2}\right)=-\cos x \). This means we need to show that the sine of the angle \( x \) shifted by \( -\frac{\pi}{2} \) is the negative of the cosine of \( x \).

Step 2: Express Sine in Terms of Cosine and Sine Using Angle Subtraction Formula

Use the sine angle subtraction formula: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \). Here, set \( a = x \) and \( b = \frac{\pi}{2} \). Substituting these into the formula, we get: \( \sin\left(x - \frac{\pi}{2}\right) = \sin x \cos\left(\frac{\pi}{2}\right) - \cos x \sin\left(\frac{\pi}{2}\right) \).

Step 3: Evaluate Trigonometric Values

Evaluate the cosine and sine of \( \frac{\pi}{2} \):\[ \cos\left(\frac{\pi}{2}\right) = 0 \quad \text{and} \quad \sin\left(\frac{\pi}{2}\right) = 1 \]

Step 4: Substitute Known Values

Substitute the known values into the expression obtained in Step 2:\[ \sin\left(x - \frac{\pi}{2}\right) = \sin x \, (0) - \cos x \, (1) \]This simplifies to:\[ \sin\left(x - \frac{\pi}{2}\right) = - \cos x \]

Step 5: Verify the Result

The result \( \sin\left(x - \frac{\pi}{2}\right) = - \cos x \) matches the identity we are proving. Therefore, the identity is correct and proven.

Key Concepts

sine angle subtraction

The sine angle subtraction formula is an important trigonometric identity that helps us determine the sine of a difference between two angles. This formula can be written as:\[ \sin(a - b) = \sin a \cos b - \cos a \sin b \]This formula allows us to break down complex trigonometric expressions by expressing the sine of angle differences using the sines and cosines of the individual angles. This breakdown simplifies the process of proving identities or solving problems involving angle subtraction.

• In our problem, we need to prove that the expression \( \sin(x - \frac{\pi}{2}) \) is equivalent to \(-\cos x\).
• We can substitute \(a = x\) and \(b = \frac{\pi}{2}\) into the general formula to apply it correctly.

The careful application of this formula leads directly to proving the original trigonometric identity, making it a powerful tool in trigonometry.

cosine function

The cosine function, often symbolized as \( \cos \), is one of the fundamental functions in trigonometry. It relates an angle in a right triangle to the ratio of the adjacent side and the hypotenuse.Cosine functions have some basic characteristics:

• The function is periodic with a period of \(2\pi\), which means it repeats its values every \(2\pi\) radians.
• The range of the cosine function is from -1 to 1 inclusive. That's because these are the extreme values that the adjacent-hypotenuse ratio can have.
• At the angle \( \frac{\pi}{2} \), the cosine value is 0.
Understanding these properties helps us solve trigonometric identities and equations. In our specific exercise, knowing \( \cos \left( \frac{\pi}{2} \right) = 0 \) played a critical role in simplifying the trigonometric identity and proving it correct. It’s this insight into the cosine function’s values that enhances our ability to manipulate and understand trigonometric expressions effectively.

angle subtraction formula

The angle subtraction formula is a set of identities in trigonometry that allow us to calculate the sine, cosine, and tangent of two angles subtracted from each other. Knowing these formulas provides a reliable way to manage expressions where the angle changes.The sine and cosine angle subtraction formulas are:

• Sine: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \)
• Cosine: \( \cos(a - b) = \cos a \cos b + \sin a \sin b \)

For the given problem, the sine subtraction formula was crucial in breaking down the identity we needed to prove. These formulas are essential tools because they transform angle subtraction problems into simple calculations using known values of trigonometric functions at particular angles. Additionally, understanding such identities allows us to expand complex expressions into manageable parts, furthering our understanding and problem-solving ability in trigonometry.
Applying these formulas gives us leverage to prove and verify the expressions quickly.