Problem 24 - Algebra and Trigonometry

Question

\(23-40\) . Prove the identity. $$ \cos \left(x-\frac{\pi}{2}\right)=\sin x $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We need to prove the trigonometric identity: \( \cos \left( x - \frac{\pi}{2} \right) = \sin x \). This means we have to show that the cosine of an angle subtracted from \( \frac{\pi}{2} \) is equal to the sine of the original angle \( x \).

Step 2: Recall Trigonometric Identities

We use the angle subtraction identity for cosine: \( \cos(a - b) = \cos a \cos b + \sin a \sin b \). Here, \( a = x \) and \( b = \frac{\pi}{2} \).

Step 3: Substitute Values into the Identity

Substitute \( \cos(\frac{\pi}{2})=0 \) and \( \sin(\frac{\pi}{2})=1 \) into the formula. Thus, we have:\[\cos \left( x - \frac{\pi}{2} \right) = \cos x \cdot 0 + \sin x \cdot 1\]

Step 4: Simplify the Expression

When we simplify the above expression, we get:\[\cos \left( x - \frac{\pi}{2} \right) = 0 + \sin x \, = \, \sin x\]

Step 5: Conclusion

The simplified equation \( \cos \left( x - \frac{\pi}{2} \right) = \sin x \) confirms the given trigonometric identity.

Key Concepts

Cosine Function

The cosine function, often denoted as \( \cos \), is one of the primary trigonometric functions. It measures the horizontal component of an angle in the unit circle. When you have an angle \( x \), \( \cos x \) represents the length from the origin to the point on the unit circle along the x-axis.

The cosine function is periodic with a period of \( 2\pi \), meaning after every \( 2\pi \) radians, the values of cosine repeat. This periodicity is crucial when solving trigonometric equations and identities.

• Cosine is even, meaning \( \cos(-x) = \cos x \).
• It starts from 1 at angle \( 0 \), decreasing to 0 at \( \frac{\pi}{2} \), reaching -1 at \( \pi \) and completing the cycle twice across \( 2\pi \).

The cosine function helps to verify and prove various trigonometric identities, including angle subtraction, used in the problem above.

Sine Function

The sine function, expressed as \( \sin \), is another fundamental trigonometric function, measuring the vertical component of an angle in the unit circle. For an angle \( x \), \( \sin x \) designates the length along the y-axis from the origin to the point on the unit circle.

Much like cosine, sine is periodic but with a period of \( 2\pi \). It has the following properties:

• Sine is an odd function: \( \sin(-x) = -\sin x \)
• It starts from 0 at angle \( 0 \), increases to 1 at \( \frac{\pi}{2} \), decreases to 0 at \( \pi \), and reaches -1 at \( \frac{3\pi}{2} \).

Understanding sine and its values at key angles, such as \( \frac{\pi}{2} \), is essential, particularly shown by the identity \( \cos(x - \frac{\pi}{2}) = \sin x \). Knowing this relationship aids in solving complex trigonometric problems by translating between sine and cosine.

Angle Subtraction Identity

The angle subtraction identity for cosine is a powerful tool in trigonometry. It states that for any angles \( a \) and \( b \), \( \cos(a - b) = \cos a \cos b + \sin a \sin b \).

This identity allows us to break down the cosine of a difference of angles into a sum involving products of sines and cosines of the individual angles.

• In the given exercise, we apply the identity with \( a = x \) and \( b = \frac{\pi}{2} \) which transforms \( \cos(x - \frac{\pi}{2}) \) to \( \cos x \cdot 0 + \sin x \cdot 1 \).
• Such transformations make it easier to simplify expressions and verify identities, as seen in proving \( \cos(x - \frac{\pi}{2}) = \sin x \) by substituting known values of \( \cos \frac{\pi}{2} \) and \( \sin \frac{\pi}{2} \).

Hence, the understanding and correct application of the angle subtraction identity is foundational to mastering trigonometric identities and relationships.