Problem 104 - Algebra and Trigonometry

Question

Verify the identity: $$ \sin \left(\frac{3 \pi}{2}-x\right)=-\cos x $$

Step-by-Step Solution

Answer

After using the identities it can be verified, \(\sin \left(\frac{3 \pi}{2}-x\right)=-\cos x\) indeed holds true.

Step 1: Using Co-Function Identity

Start by turning the given expression into a better workable expression using co-function identity:\[\sin \left(\frac{3 \pi}{2}-x\right) = \cos \left(\frac{\pi}{2} - \left(\frac{3 \pi}{2}-x\right)\right)\]

Step 2: Simplifying Inside the Cosine Function

Now, simplify the expression inside the cosine function:\[\cos \left(\frac{\pi}{2} - \left(\frac{3 \pi}{2}-x\right)\right) = \cos \left(\frac{\pi}{2} - \frac{3 \pi}{2} + x\right)\]This simplifies further to:\[\cos \left(-\pi + x\right)\]

Step 3: Using Cosine is an Even Function Rule

As cosine is an even function, cos(-x) = cos(x) holds. Therefore,\[\cos \left(-\pi + x\right) = -\cos(x)\]which completes the verification.

Key Concepts

Co-Function Identity

Understanding the co-function identity is crucial in trigonometry. A co-function identity relates trigonometric functions of complementary angles. For sine and cosine, these identities are particularly useful since they convert angles using

• co-function identities like: \( \sin(\frac{\pi}{2} - x) = \cos(x) \) and \( \cos(\frac{\pi}{2} - x) = \sin(x) \).

You can think of complementary angles as those which add up to \( \frac{\pi}{2} \) radians, or 90 degrees.
Why are these identities handy? Well, they allow the transformation of angles in functions. In the given problem, you first recognize that you can use the co-function identity to convert\[\sin \left(\frac{3 \pi}{2} - x\right) \]into a cosine function. This step simplifies the expression for easier manipulation later on in the problem.

Cosine Function

The cosine function is one of the core trigonometric functions. It expands relations between angles and side lengths in right triangles. A few key points about cosine are:

• Cosine measures the adjacent side over the hypotenuse in a right angle triangle.
• It is often denoted as \( \cos(x) \).
• Cosine also plays a pivotal role in circle-related calculations, linking back to the unit circle understanding.

In the problem, replacing sine with cosine using the identity\[\cos \left(\frac{\pi}{2} - \left(\frac{3 \pi}{2} - x\right)\right)\]allows for straightforward simplification of the angle. Understanding these replacements and simplifications can sharpen problem-solving skills when dealing with trigonometric identities.

Even Function

An even function has symmetrical properties around the y-axis. For even functions, the formula \( f(-x) = f(x) \) holds true. This characteristic reveals that these functions do not change if you input negative values. The cosine function is famously even.
In this trigonometric equation, when you reach the step\[\cos \left(-\pi + x\right) \]you can apply the evenness property of cosine: since cosine is even, we find\( \cos(-a) = \cos(a) \).
Thus,\[\cos \left(-\pi + x\right) = -\cos(x)\]proves the identity asked in the exercise. Knowing that cosine is an even function helps recognize symmetry in the problem and simplifies the expression's evaluation. Being comfortable with these symmetry properties can provide clarity when tackling complex trigonometric problems.