Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \frac{\sin t}{1-\cos t}=\csc t+\cot t $$

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to \( \frac{1+\cos t}{\sin t} \).

Step 1: Express Cosecant and Cotangent in Terms of Sine and Cosine

Recall that \( \csc t = \frac{1}{\sin t} \) and \( \cot t = \frac{\cos t}{\sin t} \). Substitute these into the right-hand side of the identity. The equation becomes: \[ \csc t + \cot t = \frac{1}{\sin t} + \frac{\cos t}{\sin t}. \]

Step 2: Combine the Right Side Expressions

Notice that both terms on the right-hand side have a common denominator \( \sin t \). Combine them: \[ \csc t + \cot t = \frac{1 + \cos t}{\sin t}. \]

Step 3: Simplify the Right Side Expression

Now, take a look at the expression you just found and compare it with the left side, which is \( \frac{\sin t}{1-\cos t} \). To prove the identity we need the right side to match the left side's expression. Consider multiplying by the conjugate \( \frac{1-\cos t}{1-\cos t} \) to simplify \( \frac{1+\cos t}{\sin t} \) further.

Step 4: Rationalize the Original Left Side

Multiply the left side \( \frac{\sin t}{1-\cos t} \) by \( \frac{1+\cos t}{1+\cos t} \), which is a form of 1 and won't change the value. This operation gives: \[ \frac{\sin t (1+\cos t)}{(1-\cos t)(1+\cos t)} = \frac{\sin t (1+\cos t)}{1-\cos^2 t}. \]

Step 5: Recognize the Pythagorean Identity

Recall the Pythagorean identity \(1-\cos^2 t=\sin^2 t\). Substitute to find: \[ \frac{\sin t (1+\cos t)}{\sin^2 t}. \]

Step 6: Simplify the Entire Expression

Cancel \( \sin t \) from numerator and denominator which results in \( \frac{1+\cos t}{\sin t} \). The left and right sides are now equal, confirming the identity.

Key Concepts

sine and cosine

Sine and cosine are fundamental trigonometric functions. They are crucial for understanding trigonometric identities, which are equations involving these functions that are true for every value of the variable. In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse. Meanwhile, cosine is the ratio of the length of the adjacent side to the hypotenuse.

In mathematical terms, we express these as:

• \( ext{sin } heta = \frac{ ext{opposite}}{ ext{hypotenuse}}\)
• \( ext{cos } heta = \frac{ ext{adjacent}}{ ext{hypotenuse}}\)

These functions are not just restricted to angles in a triangle. They extend to the unit circle, which provides values for sine and cosine for all angles, making them periodic functions with a cycle of 360° or 2\( \pi \) radians. Understanding sine and cosine is the first step toward verifying trigonometric identities. Be comfortable with their ratios and how they relate to each other.

cosecant and cotangent

Cosecant and cotangent are lesser-known, yet essential, trigonometric functions derived from sine and cosine. Knowing these functions helps in understanding and verifying complex trigonometric identities.

The cosecant function (csc) is the reciprocal of sine, which means:

• \( ext{csc } heta = \frac{1}{ ext{sin } heta}\)

On the other hand, the cotangent function (cot) is the reciprocal of tangent and also relates cosine to sine:

• \( ext{cot } heta = \frac{ ext{cos } heta}{ ext{sin } heta}\)

In verifying identities, you will often rewrite expressions using these reciprocal identities. This helps simplify complex fractions or expressions by making their components more recognizable. As seen in the example identity, expressing complex terms in terms of sine and cosine clarifies and simplifies the verification process.

Pythagorean identity

One of the most powerful tools in trigonometry is the Pythagorean identity. It connects sine and cosine in a simple yet profound equation:

\( ext{sin}^2 heta + ext{cos}^2 heta = 1\)
This identity arises from the geometry of the unit circle, where the hypotenuse is always 1, forming a right triangle with the legs represented by sine and cosine of an angle.

It is essential for simplifying expressions and proving identities, as it provides a way to replace or eliminate one of the trigonometric functions in an equation. As seen in Step 5 of the solution, recognizing that \( 1 - ext{cos}^2 t = ext{sin}^2 t \) allows for significant simplification. This swap is crucial for reducing the complexity of expressions and achieving an equivalence that confirms identities.

Mastering the Pythagorean identity gives you a powerful lens through which you can approach various trigonometric problems effectively.