Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Write expression as a single trigonometric function or a power of a trigonometric function. (You may wish to use a graph to support your result.) $$\tan \theta \cos \theta$$

Step-by-Step Solution

Answer

The expression \( \tan \theta \cos \theta \) simplifies to \( \sin \theta \).

Step 1: Identify the Expression

The given expression is \( \tan \theta \cos \theta \). Our task is to express this as a single trigonometric function or a power of a trigonometric function.

Step 2: Substitute Tangent with Sine and Cosine

Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substitute this into the expression:\[ \tan \theta \cos \theta = \frac{\sin \theta}{\cos \theta} \cdot \cos \theta \]

Step 3: Simplify the Expression

Cancel the \( \cos \theta \) terms in the expression:\[ \frac{\sin \theta}{\cos \theta} \cdot \cos \theta = \sin \theta \]

Step 4: Final Expression

After simplification, the expression \( \tan \theta \cos \theta \) reduces to \( \sin \theta \).

Key Concepts

Tangent to Sine and Cosine Conversion

Trigonometric identities allow us to interchange one trigonometric function for another, often to simplify expressions. When dealing with the tangent of an angle, it's crucial to remember that it can be expressed using sine and cosine. The identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) is fundamental here. This means that the tangent of any angle \( \theta \) is the ratio of the sine of \( \theta \) to the cosine of \( \theta \).
By using this identity, we transform expressions involving tangent into expressions with sine and cosine, which are often simpler to manipulate. For instance, in the expression \( \tan \theta \cos \theta \), substituting \( \frac{\sin \theta}{\cos \theta} \) for \( \tan \theta \) helps in simplifying the expression further, as demonstrated in the exercise.

Trigonometric Expression Simplification

Simplifying trigonometric expressions is a fundamental skill in trigonometry. It involves reducing expressions to their simplest form, which often makes them easier to interpret and use.
For the given expression \( \tan \theta \cos \theta \), once we replace \( \tan \theta \) with \( \frac{\sin \theta}{\cos \theta} \), it becomes clear that we can cancel out the \( \cos \theta \) in the numerator and the denominator. This is a straightforward process of simplification because it involves basic algebraic manipulation.

• Start with the expression \( \frac{\sin \theta}{\cos \theta} \times \cos \theta \).
• Notice that the \( \cos \theta \) in the numerator and denominator cancel each other out.
• This leaves us just \( \sin \theta \).

Thus, the expression simplifies neatly into a single trigonometric function, \( \sin \theta \), showing the power of using identities and algebraic techniques in trigonometry.

Sine Function

The sine function, represented as \( \sin \theta \), is one of the primary trigonometric functions. It describes the y-coordinate of a point on the unit circle corresponding to the angle \( \theta \). Understanding this function is crucial in trigonometry because it appears frequently in expressions and equations.
The sine function has a range of values from \(-1\) to \(1\) and varies periodically with a period of \(2\pi\).

• It starts at 0 when \( \theta \) is 0.
• It reaches its peak at 1 when \( \theta = \frac{\pi}{2} \).
• It returns to 0 at \( \pi \) and drops to -1 at \( \frac{3\pi}{2} \).
• Completes the cycle back to 0 once \( \theta \) is \( 2\pi \).

When the expression \( \tan \theta \cos \theta \) simplifies to \( \sin \theta \), it shows how deeply integrated these functions are, and how manipulating them correctly brings out familiar forms that are essential to solving trigonometric problems.