Problem 22 - Algebra and Trigonometry

Question

\(19-22\) . Prove the cofunction identity using the addition and subtraction formulas. $$ \csc \left(\frac{\pi}{2}-u\right)=\sec u $$

Step-by-Step Solution

Answer

The identity \( \csc\left(\frac{\pi}{2} - u\right) = \sec u \) is proven using complementary angles and reciprocal identities.

Step 1: Recall Cofunction Identities

The given identity to prove is \( \csc\left(\frac{\pi}{2} - u\right) = \sec u \). Cofunction identities relate the trigonometric functions of complementary angles, and recognizing these identities can be key in solving the problem.

Step 2: Use Cosecant Definition with Sine

Cosecant is the reciprocal of sine, so we start by expressing cosecant in terms of sine: \( \csc\left(\frac{\pi}{2} - u\right) = \frac{1}{\sin\left(\frac{\pi}{2} - u\right)} \).

Step 3: Apply the Sine Complementary Angle Identity

Use the identity for sine of a complementary angle: \( \sin\left(\frac{\pi}{2} - u\right) = \cos u \). Substitute this into the expression: \( \csc\left(\frac{\pi}{2} - u\right) = \frac{1}{\cos u} \).

Step 4: Recognize Reciprocal Identity of Cosine

By definition, the reciprocal of cosine is secant. Therefore, \( \frac{1}{\cos u} \equiv \sec u \). Thus, we have \( \csc\left(\frac{\pi}{2} - u\right) = \sec u \).

Step 5: Final Verification

Ensure that the steps logically follow and make sense: starting from the cofunction identity for sine, transitioning from \( \sin\left(\frac{\pi}{2} - u\right) = \cos u \), and using the reciprocal relationship to match with the secant function.

Key Concepts

Addition and Subtraction Formulas

The addition and subtraction formulas are essential tools in trigonometry, helping you to find the sine, cosine, or tangent of two angles combined or subtracted. For example, the sine addition formula can be expressed as

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

Similarly, the sine subtraction formula is:

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

These formulas are not only useful for simplifying expressions but also for proving more complicated identities. In this context, they help us manipulate and solve problems involving unknown angle measures.
The power of these formulas lies in breaking down complex expressions into simpler, more workable components. When tackling a problem involving trigonometric identities or cofunction relations, knowing these formulas allows us to see relationships between angles differently. This helps solve the problem by breaking it into smaller, more manageable parts.

Cosecant Reciprocal Identity

The reciprocal identity is fundamental in understanding the relationship between different trigonometric functions. For the cosecant, it is defined as the reciprocal of the sine function, expressed as:

• \( \csc(\theta) = \frac{1}{\sin(\theta)} \)

In the given problem, this identity allows us to express the cosecant in terms of sine, which is a crucial step to proving the provided identity.
This transformation from cosecant to sine helps simplify the expression, providing a path towards solving the equation more easily. The goal is to substitute the function with its equivalent reciprocal to make it easier to recognize and apply additional identities, such as those involving complementary angles. This step is a good illustration of how a single identity can transform the problem, making it more approachable and solvable.

Complementary Angles

Complementary angles have a special property in trigonometry: their measures add up to \( \frac{\pi}{2} \) radians (or 90 degrees). Important cofunction identities rely on this relationship because the trigonometric values of complementary angles are directly related.
In this exercise, the cofunction identity for sine comes into play as:

• \( \sin\left(\frac{\pi}{2} - u\right) = \cos u \)

The identity states that the sine of an angle's complement is the cosine of the angle itself. This mirrors the fact that every trigonometric function has a complementary cofunction identity.
This property is essential for converting between trigonometric functions of complementary angles. Understanding this connection not only assists in proving identities like the one given but also serves as a foundational concept in trigonometry, helping you to solve a variety of problems more efficiently.

Sine and Cosine Identities

Sine and cosine identities form the backbone of trigonometric problem-solving and are critical to mastering trigonometry. They provide us with methods to transform trigonometric statements into more familiar forms for easier manipulation.

• Basic identities include the famous Pythagorean identity: \( \sin^2\theta + \cos^2\theta = 1 \)

This identity is a cornerstone, helping you relate sine and cosine in many problems. Additionally, knowing that \( \sin(\frac{\pi}{2} - u) = \cos u \) and \( \cos(\frac{\pi}{2} - u) = \sin u \) helps demonstrate the bound relationship between these functions.
These identities underline the symmetry and cyclical nature of trigonometric functions, which can be genuinely enlightening. By leveraging these relationships, complex trigonometric problems become more straightforward and intuitive. This interconnected system allows you to pivot between functions effortlessly, applying each identity as a puzzle piece that fits into the overarching trigonometric picture.