Problem 52 - Contemporary Precalculus

Question

In Exercises \(49-54\), prove the given identity. \(\cot (-t)=-\cot t[\text {Hint}:\) Express the left side in terms of sine and cosine; then use the negative angle identities and express the result in terms of cotangent.]

Step-by-Step Solution

Answer

Question: Prove the identity \(cot(-t) = -cot t\). Answer: We used the formula for cotangent \(\cot(t) = \frac{\cos(t)}{\sin(t)}\) and proved the given identity by applying the negative angle identities. The final result is \(\cot(-t) = -\cot t\).

Step 1: Expressing cotangent in terms of sine and cosine

Since cotangent is the reciprocal of tangent, we can express it in terms of sine and cosine functions. $$\cot(t) = \frac{\cos(t)}{\sin(t)}$$

Step 2: Applying the negative angle identities to the left side

Replace t with -t in the cotangent formula, and then apply the negative angle identities. $$\cot(-t) = \frac{\cos(-t)}{\sin(-t)}$$ Using the identities: $$\cot(-t) = \frac{\cos t}{-\sin t}$$

Step 3: Express the result in terms of the cotangent

To prove the given identity, we need to express the result obtained in step 2 in terms of cotangent. $$\cot(-t) = -\frac{\cos t}{\sin t}$$ Recall that \(\cot(t) = \frac{\cos(t)}{\sin(t)}\), so we can replace \(\frac{\cos t}{\sin t}\) with \(\cot(t)\): $$\cot(-t) = -\cot t$$ Therefore, the identity is proven: $$\cot(-t) = -\cot t.$$

Key Concepts

Understanding Cotangent

The cotangent function is closely related to the tangent function, which is a fundamental trigonometric function. It is defined as the reciprocal of the tangent. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Thus, cotangent is the ratio of the adjacent side to the opposite side.

In terms of sine and cosine, cotangent can be expressed as the ratio of cosine to sine:

• \( \cot(t) = \frac{\cos(t)}{\sin(t)} \)

This relationship is useful because it allows us to express tangent and cotangent in terms of more familiar functions: sine and cosine.

Cotangent, like other trigonometric functions, is periodic, which means it repeats its values in regular intervals. Understanding the behavior of cotangent over its period is important in solving trigonometric equations and proving identities.

Negative Angle Identities Explained

Negative angle identities are a set of trigonometric rules that describe how trigonometric functions behave when their angle arguments are negated. These identities are particularly helpful when dealing with reflections over the origin or when simplifying expressions.

For cosine and sine, the negative angle identities are:

• \( \cos(-t) = \cos(t) \) — Cosine is an even function.
• \( \sin(-t) = -\sin(t) \) — Sine is an odd function.

These identities reveal the symmetry of the sine and cosine functions about the origin. The signs of the functions change based on the quadrant in which the angle lies. This concept is crucial for accurately simplifying and proving trigonometric identities like the one involving cotangent.

When applying these identities, one can often simplify and transform complex trigonometric expressions to work with more straightforward expressions.

The Sine and Cosine Functions

Sine and cosine are fundamental building blocks in trigonometry and are essential in defining other trigonometric functions. Understanding their properties can significantly help in solving problems.

Properties of Sine and Cosine:

• The cosine function \(\cos(t)\) is the x-coordinate of a point on the unit circle corresponding to the angle \(t\).
• The sine function \(\sin(t)\) is the y-coordinate of a point on the unit circle corresponding to the angle \(t\).
• Both sine and cosine functions have a range of values from -1 to 1.
• Sine and cosine are periodic functions with period \(2\pi\).

These functions are used to describe oscillatory motion, wave patterns, and are deeply interrelated with exponential functions in complex analysis.

By understanding these functions, one could predict the values and behaviors of the trigonometric equations and further harness these relationships in proving identities like \(\cot(-t) = -\cot t\). This ties back into the mathematical proof that evaluates cotangent through a combination of sine and cosine.