Problem 40 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \cos \theta-\sin \theta+\csc \theta=\frac{\sin \theta+\cos \theta}{\tan \theta} $$

Step-by-Step Solution

Answer

The given identity is verified.

Step 1: Simplify the Left-Hand Side (LHS)

Start by expressing the terms of the LHS in terms of sine and cosine. We have \( \cos \theta - \sin \theta + \csc \theta \). Since \( \csc \theta = \frac{1}{\sin \theta} \), the LHS becomes:\[ \cos \theta - \sin \theta + \frac{1}{\sin \theta} \]

Step 2: Simplify the Right-Hand Side (RHS)

Look at the RHS \( \frac{\sin \theta + \cos \theta}{\tan \theta} \). Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), so the RHS becomes:\[ \frac{\sin \theta + \cos \theta}{\frac{\sin \theta}{\cos \theta}} = (\sin \theta + \cos \theta) \cdot \frac{\cos \theta}{\sin \theta} \] which simplifies to:\[ \cos \theta + \frac{\cos \theta^2}{\sin \theta} \]

Step 3: Clear the Fractions of LHS

To facilitate comparison, clear the fractions in the LHS by finding a common denominator, which is \( \sin \theta \).\[ \frac{\cos \theta \sin \theta}{\sin \theta} - \frac{\sin^2 \theta}{\sin \theta} + \frac{1}{\sin \theta} = \frac{\cos \theta \sin \theta - \sin^2 \theta + 1}{\sin \theta} \]

Step 4: Equate Numerators

Now that both sides have the same denominator, equate the numerators:From LHS:\[ \cos \theta \sin \theta - \sin^2 \theta + 1 \]From RHS:\[ \cos \theta \sin \theta + \cos^2 \theta \]Recognize that \( 1 = \sin^2 \theta + \cos^2 \theta \), and substitute into LHS:\[ \cos \theta \sin \theta - \sin^2 \theta + \sin^2 \theta + \cos^2 \theta \] which simplifies to:\[ \cos \theta \sin \theta + \cos^2 \theta \]

Step 5: Conclude the Verification

Since both sides simplify to the same expression \( \cos \theta \sin \theta + \cos^2 \theta \), the identity is verified.

Key Concepts

Understanding Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions that relate an angle in a right triangle to the ratios of two of its sides. These functions play a crucial role in a variety of mathematical and scientific applications, providing the foundation for understanding wave behavior and circular motion.
* **Sine Function**: Denoted by \( \sin \theta \), it is the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle. The sine function oscillates between -1 and 1 as the angle increases from 0 to 360 degrees.* **Cosine Function**: Represented by \( \cos \theta \), it is the ratio of the length of the adjacent side to the hypotenuse in a right triangle. The cosine function also varies between -1 and 1 and is complementary to the sine function.These functions are periodic, with a period of \( 360^{\circ} \) or \( 2\pi \) radians.
Understanding these functions aids in solving trigonometric identities, like the one in the exercise, by expressing other trigonometric functions through them.

Exploring the Cosecant Function

The cosecant function is one of the reciprocal trigonometric functions and is closely related to the sine function. Understanding this function is essential when dealing with trigonometric identities that involve recipricals.
* **Cosecant Function**: Denoted by \( \csc \theta \), it is the reciprocal of the sine function, meaning \( \csc \theta = \frac{1}{\sin \theta}\). Because it is the reciprocal, \( \csc \theta \) is undefined whenever \( \sin \theta = 0 \) since division by zero is not possible. In terms of the unit circle, while the sine value represents the y-coordinate of a point on the circle, the cosecant is related to the arcs and segments that can extend beyond the unit circle if sine is zero.
The understanding of this function assists in manipulating trigonometric expressions, as seen when simplifying the left side of the original exercise.

Deciphering the Tangent Function

The tangent function, another essential trigonometric function, is derived from the sine and cosine functions. It expresses a different relationship between the sides of a right triangle.
* **Tangent Function**: Denoted as \( \tan \theta \), tangent is the ratio of sine to cosine, so \( \tan \theta = \frac{\sin \theta}{\cos \theta}\). This function expresses the relationship between the opposite and adjacent sides. * Properties of Tangent: * Undefined when \( \cos \theta = 0 \), as division by zero cannot occur. * Repeats or is periodic every \( 180^{\circ} \) or \( \pi \) radians.When simplifying trigonometric identities, converting expressions into terms of tangent can sometimes simplify the process, as demonstrated in the exercise. Recognizing the conversion and behavior of tangent helps understand how the right side of the identity equals the left when simplified properly.