Problem 144 - Calculus Volume 1

Question

For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only. $$\sin x(\csc x-\sin x)$$

Step-by-Step Solution

Answer

The simplified expression is \(\cos^2 x\).

Step 1: Rewrite the Expression in Terms of Sine and Cosine

Start with the given expression: \[ \sin x(\csc x - \sin x) \]Recall that the cosecant function, \(\csc x\), is the reciprocal of the sine function. Therefore, we can express \(\csc x\) as \(\frac{1}{\sin x}\). Substitute this into the expression:\[ \sin x \left( \frac{1}{\sin x} - \sin x \right) \]

Step 2: Distribute the Sine Function

Distribute \(\sin x\) across the terms inside the parentheses:\[= \sin x \cdot \frac{1}{\sin x} - \sin x \cdot \sin x \]

Step 3: Simplify Each Term

For the first term, \(\sin x \cdot \frac{1}{\sin x}\) simplifies to 1 because \(\sin x\) cancels out with its reciprocal.The second term, \(\sin x \cdot \sin x\), is simply \(\sin^2 x\). Thus, the expression becomes:\[ = 1 - \sin^2 x \]

Step 4: Final Simplification Using Pythagorean Identity

Recall the Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \). Solve for \( \cos^2 x \) by rearranging the identity:\[ \cos^2 x = 1 - \sin^2 x \]Substitute this back into the expression:\[ 1 - \sin^2 x = \cos^2 x \]

Key Concepts

sine and cosine

Sine and cosine are fundamental trigonometric functions applicable in various mathematics fields, including algebra, calculus, and geometry. They primarily arise from studying right triangles but extend to the unit circle to define periodic behavior in waves.
Sine, abbreviated as \( \sin \), measures the ratio of the opposite side to the hypotenuse in a right triangle. Similarly, cosine, abbreviated as \( \cos \), represents the adjacent side to the hypotenuse ratio in a right triangle.

• Unit Circle Connection: On the unit circle, sine corresponds to the y-coordinate, and cosine to the x-coordinate, of a point on the circle.
• Applications: These functions help model periodic phenomena like sound and light waves, and oscillations.
• Properties: Both sine and cosine are periodic with a period of \( 2\pi \) and are bounded between -1 and 1.

Due to their complementary nature, sine and cosine often appear in equations and identities together. Understanding these basics is vital for delving deeper into trigonometric identities.

cosecant function

The cosecant function, denoted as \( \csc x \), is one of the reciprocal trigonometric functions. Specifically, it is the reciprocal of the sine function.
The formula for the cosecant is: \[ \csc x = \frac{1}{\sin x} \]
This means that the cosecant of an angle in a right triangle is the ratio of the hypotenuse to the opposite side.

• Domain and Range: \( \csc x \) is undefined wherever \( \sin x = 0 \). Thus, it excludes multiples of \( \pi \).
• Cosecant and Unit Circle: Like sine, cosecant derives its values from the unit circle, but it inverts them.
• Graph: The graph of \( \csc x \) resembles flipped sine waves with vertical asymptotes where sine is zero.

Understanding \( \csc x \) is crucial, especially when simplifying trigonometric expressions, as seen in rewriting functions in terms of sines and cosines.

Pythagorean identity

The Pythagorean identity is one of the most significant trig identities, forming the backbone of many simplifications. The core form of the identity is expressed as \[ \sin^2 x + \cos^2 x = 1 \]
This identity stems from Pythagoras' theorem in a right triangle when applied to the unit circle.

• Variants: Rearranging the identity provides useful forms like \( \cos^2 x = 1 - \sin^2 x \) and \( \sin^2 x = 1 - \cos^2 x \).
• Use in Simplification: We frequently use the identity to replace squares of sine or cosine with the other function.
• Visualization: Visualize this identity as being derived from the unit circle, ensuring the sum of squares of the coordinates equals one.

By utilizing the Pythagorean identity, complex trigonometric expressions simplify greatly, reducing them to more manageable forms, thus highlighting their importance in trigonometry.