Problem 99 - Precalculus : Building Concepts and Connections

Question

This set of exercises will draw on the ideas presented in this section and your general math background. $$\text { Show that } \csc \left(90^{\circ}-\theta\right)=\sec \theta$$

Step-by-Step Solution

Answer

\(\csc(90^{\circ}-\theta) = \sec(\theta)\) is true for all values of angle \(\theta\) as both sides of the equation are equal.

Step 1: Jot down the identities

Firstly, remember the basic identities: \(\csc(\theta) = 1 / \sin(\theta)\) and \(\sec(\theta) = 1 / \cos(\theta)\). Secondly, recall that the sine of a complementary angle is equal to the cosine of the angle, and the cosine of a complementary angle equals the sine of the angle.

Step 2: Apply the identities

Applying these identities to the problem at hand, the left hand side of the equation becomes \(\csc(90^{\circ}-\theta) = 1 / \sin(90^{\circ} - \theta)\) which can be simplified to \(1 / \cos(\theta)\) by using the identity that \(\sin(90^{\circ} - \theta) = \cos(\theta)\). This implies that \(\csc(90^{\circ}-\theta)=\sec(\theta)\), as \(1 / \cos(\theta) = \sec(\theta)\).

Step 3: Confirm the result

Now comparing both sides of the given equation, they are the same, proving that \(\csc(90^{\circ}-\theta) = \sec(\theta)\).

Key Concepts

Cosecant Function

The cosecant function, denoted as \(\csc\), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the sine function. In other words, for any angle \(\theta\), the cosecant of \(\theta\) is given by \(\csc(\theta) = \frac{1}{\sin(\theta)}\).

Understanding the cosecant function is crucial when dealing with right-angled triangles and waves, as it provides a direct relationship with the sine function, which is commonly associated with these applications. Moreover, \(\csc\) is undefined when \(\sin(\theta)\) is zero, corresponding to angles where the opposite side of a right-angled triangle would have zero length, which is not possible in real-world scenarios.

When you encounter \(\csc\) in exercises, it's essential to remember its connection to the sine function and that it can help you find the length of the hypotenuse when given the length of the opposite side in a right-angled triangle.

Secant Function

Similar to the cosecant function, the secant function, denoted as \(\sec\), is another fundamental trigonometric function and is defined as the reciprocal of the cosine function. For any angle \(\theta\), the secant of \(\theta\) is represented as \(\sec(\theta) = \frac{1}{\cos(\theta)}\).

In real-life applications, the secant function appears in fields such as physics, engineering, and astronomy, wherever waves or oscillations are discussed, or in architectural design when determining angles in structures.

Understanding that \(\sec\) is undefined for the angles where \(\cos(\theta)\) equals zero is essential because those would imply a division by zero which is mathematically undefined. The secant function, similar to cosecant, helps map the relationship between the length of the hypotenuse and the adjacent side in right-angled triangles.

Complementary Angles

Complementary angles are two angles that add up to \(90^\circ\) or \(\frac{\pi}{2}\) radians. In the context of trigonometry, complementary angles have a unique relationship: the sine of one angle is equal to the cosine of its complement, and vice versa; symbolically, \(\sin(90^\circ - \theta) = \cos(\theta)\) and \(\cos(90^\circ - \theta) = \sin(\theta)\).

This property is extremely useful when solving problems involving right angles and when working with pairs of acute angles in right-angled triangles. It allows us to simplify expressions and solve equations involving trigonometric functions by recognizing that an angle and its complement have their sine and cosine values 'swapped'.

When working with complementary angles, it's also important to note that this relationship only holds in a right-angle geometric context, which has foundational implications on how angles interact within trigonometry.