Problem 42 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The identity \( \sin \left(\frac{3 \pi}{2}-x\right)=-\cos x \) has been confirmed to be true.

Step 1: Replace \(\sin\) argument

For any angle \(\alpha\), the sin function can be rewritten as \(sin (\pi/2 - \alpha) = cos \alpha\). Therefore, replace \(\sin \left(\frac{3 \pi}{2}-x\right)\) using this identity to get \(cos(\pi - x)\).

Step 2: Rewrite cosine argument

The cos function has the property that \(cos (\pi - \alpha) = - cos \alpha\). Therefore, replace \(cos(\pi - x)\) using this property to get \(-cos x\).

Step 3: Check the Equivalence

Verify the equivalence, ensuring that \(-cos x = -cos x\). This confirms that the trigonometric identity given in the problem holds true.

Key Concepts

sin and cos relationships

The relationship between sine and cosine is one of the foundational aspects of trigonometry. They are closely linked due to their complementary angles. In the unit circle, sine and cosine represent the y-coordinate and x-coordinate, respectively, of a point. This relationship can change expressions into equivalent forms easily.

For example, the identity \( \sin(\pi/2 - \alpha) = \cos \alpha \) is a classic demonstration of this relationship. Essentially, it states that the sine of an angle is equal to the cosine of its complement. This formula is incredibly useful and shows just how interconnected these functions are.

• Both functions are periodic, with a period of \( 2\pi \).
• The range for sine and cosine is from -1 to 1.
• They maintain symmetry: sine is odd while cosine is even.

Understanding these relationships allows for simplifications and transformations in trigonometric equations. It’s why rearranging and substituting terms in an identity can help match one side with the other, as seen in the problem.

angle transformations

Trigonometric identities often involve transformations of angles which allow us to simplify expressions. These transformations are based on known identities. For example, angles can be manipulated through addition or subtraction formulas.

A useful transformation used in the original exercise was substituting the angle in the sine function. We changed \( \sin(\frac{3\pi}{2} - x) \) into \( \cos(\pi - x) \). This was achieved using known transformations such as:

• \( \sin(\pi/2 - \alpha) = \cos \alpha \)
• \( \cos(\pi - \alpha) = -\cos \alpha \)

Such transformations make it possible to convert into simpler expressions. It allows one to unveil equalities that might not seem obvious at the start.
An important aspect is to remember that angle transformations require precise application. Ensure that each step aligns with trigonometric identities already established.

verifying identities

Verifying trigonometric identities involves proving one side of an equation equals the other through logical steps and transformations. This requires applying known identities and transformations smartly.
The process usually involves:

• Choosing the more complex side to simplify.
• Using identities like Pythagorean or angle transformations to simplify or transform terms.
• Working step-by-step to achieve equivalence.

In the given problem, we verified that \( \sin(\frac{3\pi}{2}-x) = -\cos x \) by converting the sine term into cosine expressions and confirming equivalence. It showed the power of identities to simplify expressions into recognizable forms.
Remember, proficiency in verifying identities comes with understanding various trigonometric properties and practicing their applications.
Ultimately, being able to verify identities helps solve trigonometric equations with confidence, as it trains both attention to detail and problem-solving skills.