Problem 140 - Calculus Volume 1

Question

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. \(\sec x \sin x \cot x\)

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Rewrite Secant and Cotangent in Terms of Sine and Cosine

The goal is to express everything using sines and cosines. \( \sec x \) is the reciprocal of cosine, so \( \sec x = \frac{1}{\cos x} \). \( \cot x \) is the reciprocal of tangent, so \( \cot x = \frac{\cos x}{\sin x} \). We can rewrite the expression \( \sec x \sin x \cot x = \left( \frac{1}{\cos x} \right) \sin x \left( \frac{\cos x}{\sin x} \right) \).

Step 2: Simplify the Expression

Now that all terms are expressed in sines and cosines, we can simplify: \( \left( \frac{1}{\cos x} \right) \sin x \left( \frac{\cos x}{\sin x} \right) = \frac{\sin x}{\cos x} \times \frac{\cos x}{\sin x} \). The \( \sin x \) terms in the numerator and denominator cancel out, as do the \( \cos x \) terms, leaving \( 1 \).

Step 3: Write the Simplified Expression

After cancellation, the simplified expression is \( 1 \). There is no need to express \( 1 \) further in terms of sine and cosine.

Key Concepts

Sine

The sine function is one of the fundamental trigonometric functions, commonly abbreviated as "sin." It relates the angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse. This can be expressed as:\[\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\]Sine is often used because of its periodic nature, which means it repeats its values in a regular pattern over intervals. This periodicity is particularly useful in various fields such as physics, engineering, and graphics.

• The sine function varies smoothly and continuously over its domain, which is all real numbers.
• The range of the sine function is between -1 and 1, inclusive.
• It achieves its maximum value of 1 at certain angles like \(90^\circ\) or \(\frac{\pi}{2}\) radians, and its minimum value of -1 at angles like \(270^\circ\) or \(\frac{3\pi}{2}\) radians.

Understanding sine is crucial as it is often used in various trigonometric identities, such as simplifying complex trigonometric expressions.

Cosine

Much like sine, the cosine function is a key component in trigonometry, abbreviated as "cos." The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. This is demonstrated by:\[\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\]The cosine function shares many properties with the sine function, such as periodicity:

• Cosine is also periodic and repeats its values regularly, with a complete cycle over \(360^\circ\) or \(2\pi\) radians.
• The function's range is the same as sine, between -1 and 1.
• Cosine peaks at 1 at angles like \(0^\circ\) or \(0\pi\) radians, and sinks to -1 at \(180^\circ\) or \(\pi\) radians.

Cosine is highly valuable in understanding and applying trigonometric identities, which rely on these periodic properties for simplification and problem-solving tasks.

Reciprocal Identities

Reciprocal identities are a set of relationships in trigonometry where each basic trigonometric function has a reciprocal counterpart. These are particularly handy when simplifying expressions or solving equations.For clarity:

• The reciprocal of sine is the cosecant function, expressed as \( \csc x = \frac{1}{\sin x} \).
• The reciprocal of cosine is the secant function, written \( \sec x = \frac{1}{\cos x} \).
• Lastly, the reciprocal of the tangent function is cotangent, noted as \( \cot x = \frac{1}{\tan x} \).

These identities help transform trigonometric expressions into simpler or more useful forms, as shown in the original exercise solution. By converting functions such as secant and cotangent into their reciprocal sine and cosine terms, the original complex expression reduces significantly, making calculations more straightforward.