Problem 45 - Precalculus with Limits

Question

In Exercises 9-50, verify the identity \( \sin t \csc \left(\dfrac{\pi}{2} - t \right) = \tan t \)

Step-by-Step Solution

Answer

As the left hand side of the equation simplifies to \(\tan t \), which is equal to the right hand side of the original equation, the identity \(\sin t \csc \left(\dfrac{\pi}{2} - t \right) = \tan t \) is verified.

Step 1: Write down known Identity

The given identity to verify is \(\sin t \csc \left(\dfrac{\pi}{2} - t \right) = \tan t \). The key to solving trigonometric identities is knowing the relationships between different trigonometric functions. A crucial identity that will be helpful here is: \(\csc \theta = \frac{1}{\sin \theta} \)

Step 2: Replace Cosecant

Replace \(\csc \left(\dfrac{\pi}{2} - t \right)\) with \(\frac{1}{\sin \left(\dfrac{\pi}{2} - t \right)}\) on the LHS to get \(\sin t \cdot \frac{1}{\sin \left(\dfrac{\pi}{2} - t \right)}\).

Step 3: Apply Complementary Angle formula

In trigonometry, the complementary angle formula can be applied which tells that \(\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta\). So the sin in the denominator can be replaced to get \(\sin t \cdot \frac{1}{\cos t}\).

Step 4: Simplify the expression

The expression \(\sin x / \cos x\) is equivalent to \(\tan x\). Therefore, the left hand side of the initial equation simplifies down to \(\tan t\).

Key Concepts

Complementary Angle

When working with trigonometric functions, the concept of complementary angles is essential. In trigonometry, two angles are complementary if their sum equals 90 degrees or \( \frac{\pi}{2} \) radians. This relationship leads to some useful identities involving trigonometric functions.
Consider the function \( \sin \left( \frac{\pi}{2} - \theta \right) \). According to the complementary angle identity, this is equivalent to \( \cos \theta \). This means that the sine of an angle's complement is the cosine of the angle itself.

• For example, if \( \theta = 30^\circ \), then \( \sin(60^\circ) = \cos(30^\circ) \).

Other functions, like tangent, also relate through complementary angles. Knowing these identities helps simplify complex expressions, making it easier to verify trigonometric equations. These identities are particularly helpful when converting one trigonometric function into another which might be more suitable for the problem at hand.

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is the reciprocal of the sine function. Given a particular angle \( \theta \), the formula is:

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

This function arises whenever you need to "invert" the sine in mathematical expressions and is particularly useful for simplifying certain identities.
If you have an equation involving \( \csc \left( \frac{\pi}{2} - t \right) \), you can easily rewrite this using the sine of the complementary angle. As shown in the identity \( \csc \theta = \frac{1}{\sin \theta} \), substituting \( \sin \left( \frac{\pi}{2} - t \right) \) with \( \cos t \) due to the complementary angle theorem, further simplifies the expression.
Cosecant, like other reciprocal functions such as secant and cotangent, often appears less frequently but holds the same importance as other basic trigonometric functions in developing and verifying identities.

Tangent Function

The tangent function is one of the fundamental trigonometric functions and is expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This ratio highlights the relationship between the sine and cosine of the same angle, crucial for solving many trigonometric problems.
The identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) was a key factor in the verification process in the given exercise. By expressing the left side of the initial trigonometric equation in terms of sine and cosine, it becomes clear that the simplified result is indeed \( \tan \theta \).

• This fundamental identity facilitates the transformation between different trigonometric functions.
• It allows the conversion of multiplicative expressions into simpler forms when validating or solving trigonometric equations.
• Understanding tangent as a ratio helps inform broader applications in trigonometry like solving triangles or modeling periodic phenomena.

In this way, the tangent function plays a central role in expressing and transforming trigonometric identities to achieve the desired forms in mathematical proofs and equations.