Problem 23 - Algebra 2

Question

Simplify each trigonometric expression. $$ \tan \theta(\cot \theta+\tan \theta) $$

Step-by-Step Solution

Answer

The simplified form of \( \tan \theta(\cot \theta+ \tan \theta) \) is \( \sec^2 \theta \).

Step 1: Substitute the Cotangent

In this step, substitute the cotangent with the reciprocal of the tangent, as they are reciprocals of each other. So, the expression becomes \( \tan \theta(\frac{1}{\tan \theta}+ \tan \theta)\).

Step 2: Simplify the Expression

After substituting, distribute \( \tan \theta \) through the parentheses, to get \(1+ \tan^2 \theta \). This simplifies to \( \sec^2 \theta \).

Step 3: Final step

The simplified expression is \( \sec^2 \theta \).

Key Concepts

Tangent Function

The tangent function is one of the primary trigonometric functions and is often represented as \( \tan \theta \). It relates two sides of a right triangle: the opposite side and the adjacent side relative to a given angle \( \theta \). In mathematical terms, we write it as:

• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)

In terms of sine and cosine, another common formula that describes the tangent function is:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

The tangent function is particularly useful in various branches of science and engineering due to its periodic and wave-like properties. Remember, the angle \( \theta \) could be any real number, and its value will determine the result of the function. Keep in mind, divisors in the tangent function (i.e., \( \cos \theta \)) should not be zero to maintain valid calculations.

Cotangent Function

The cotangent function is another fundamental trigonometric function. It is the reciprocal of the tangent function and is often represented as \( \cot \theta \). This means it can be calculated by taking the inverse of tangent:

• \( \cot \theta = \frac{1}{\tan \theta} \)

In terms of triangle sides, cotangent is defined as the adjacent side over the opposite side:

• \( \cot \theta = \frac{\text{adjacent}}{\text{opposite}} \)

Additionally, using sine and cosine, the cotangent can also be given by:

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

This property illustrates its relationship with other trigonometric functions and helps simplify expressions involving trigonometric identities. Just like with the tangent function, the angle \( \theta \) should ensure that \( \sin \theta \) does not equal zero to avoid undefined results.

Secant Function

The secant function is derived from the cosine function and is symbolized as \( \sec \theta \). It is defined as the reciprocal of the cosine function:

• \( \sec \theta = \frac{1}{\cos \theta} \)

This function provides insight into the concept of an angle's relationship within a unit circle, as the cosine represents the horizontal component of a point on the circle. The secant function shares important characteristics with its reciprocal (the cosine function), except it tends toward infinity wherever cosine crosses zero. This makes its graph quite different, having vertical asymptotes where \( \cos \theta \) is zero.
A vital identity involving secant is its relationship with tangent:

• \( 1 + \tan^2 \theta = \sec^2 \theta \)

This identity is frequently used to simplify complex trigonometric expressions, just as seen in the original exercise given. Paying attention to the relationship between these functions can greatly enhance problem-solving skills in trigonometry.