Problem 4 - Algebra 2

Question

Verify each identity. $$ \csc \left(\frac{\pi}{2}-\theta\right)=\sec \theta $$

Step-by-Step Solution

Answer

The given trigonometric identity \(\csc \left(\frac{\pi}{2}-\theta \right)=\sec \theta\) is verified.

Step 1: Understanding the problem

The given equation is in the form of a trigonometric identity. It needs to be proven that the left-hand side of the equation (LHS) is equal to the right-hand side (RHS). That is, \(\csc \left(\frac{\pi}{2}-\theta \right) = \sec \theta\).

Step 2: Working with the Left-Hand Side (LHS)

Start from the LHS of the equation, which is \(\csc \left(\frac{\pi}{2}-\theta \right)\). Now, use the identity \(\csc (\alpha) = 1/\sin(\alpha)\). So it can be written as \(1/\sin \left(\frac{\pi}{2}-\theta \right) \).

Step 3: Using the trigonometric identity for the complement of an angle

Next, as \(\sin(\pi/2 - \alpha) = \cos(\alpha)\). This implies \(1/\sin \left(\frac{\pi}{2}-\theta \right) = 1/\cos(\theta) \). This is equal to \(\sec(\theta)\).

Step 4: Verifying the identity

The LHS has now been transformed to the RHS, thus the initial equation has been verified: \(\csc \left(\frac{\pi}{2}-\theta \right)=\sec \theta\)

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is the reciprocal of the sine function. This means it is the inverse of sine; mathematically expressed as \( \csc \theta = \frac{1}{\sin \theta} \). When using the cosecant function, it's essential to understand that it is undefined when \( \sin \theta = 0 \). This occurs at values like \( \theta = n \pi \), where \( n \) is an integer. In practical terms:

• The cosecant function helps us find the ratio of the hypotenuse to the opposite side in a right triangle.
• It is used in various trigonometric identities and plays a critical role in solving trig equations.

Understanding \( \csc \theta \) is vital for working with complementary angles, as showcased in the initial problem, where \( \csc(\frac{\pi}{2} - \theta) \) relates closely to \( \sec \theta \). Remember that knowing when and how to apply the reciprocal relation \( \csc \left( \alpha \right) = \frac{1}{\sin(\alpha)} \) can help simplify and verify many trigonometric identities.

Secant Function

The secant function, represented as \( \sec \theta \), is the reciprocal of the cosine function. Therefore, its formula is \( \sec \theta = \frac{1}{\cos \theta} \). Like the cosecant function, the secant function is undefined when \( \cos \theta = 0 \), happening when \( \theta = \frac{\pi}{2} + n\pi \) for integer \( n \). Some key features of \( \sec \theta \):

• It provides the ratio of the hypotenuse to the adjacent side in a right triangle.
• It's crucial in solving trigonometric identities where cosine is involved.

In the original exercise, \( \sec \theta \) is the term equated to \( \csc(\frac{\pi}{2} - \theta) \). By understanding that \( \sec \theta = \frac{1}{\cos \theta} \), you see how the transformations on complementary angles lead to the trigonometric identity being verified. This connection illustrates how knowing one trigonometric ratio can help you find others.

Complementary Angles

Complementary angles are two angles whose sum equals \( \frac{\pi}{2} \) radians (or \( 90^{\circ} \)). This relationship is foundational in trigonometry since many identities transform one function into another through complementary angles.For instance:

• \( \sin(\frac{\pi}{2} - \theta) = \cos(\theta) \)
• \( \cos(\frac{\pi}{2} - \theta) = \sin(\theta) \)

These identities are crucial because they allow for the conversion between trigonometric functions using angle subtraction. In our original exercise, using complementary angle identities transformed \( \csc(\frac{\pi}{2} - \theta) \) to \( \sec \theta \), helping to verify the trigonometric identity. By understanding complementary angles, you can simplify complex trigonometric problems and understand the inherent symmetries in trigonometry.