Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
sin θ sec θ

Verified step by step guidance

1. Recall the definition of the secant function in terms of cosine: \(\sec \theta = \frac{1}{\cos \theta}\).
2. Rewrite the given expression \(\sin \theta \sec \theta\) by substituting \(\sec \theta\) with \(\frac{1}{\cos \theta}\), so it becomes \(\sin \theta \times \frac{1}{\cos \theta}\).
3. Express the product as a single fraction: \(\frac{\sin \theta}{\cos \theta}\).
4. Recognize that \(\frac{\sin \theta}{\cos \theta}\) is the definition of the tangent function, \(\tan \theta\).
5. Therefore, the expression simplifies to \(\tan \theta\), which is written only in terms of sine and cosine without any quotients involving secant.

Key Concepts

Trigonometric Functions and Their Definitions

Trigonometric functions like sine, cosine, and secant relate angles to ratios of sides in a right triangle or points on the unit circle. Sine (sin θ) is the ratio of the opposite side to the hypotenuse, cosine (cos θ) is adjacent over hypotenuse, and secant (sec θ) is the reciprocal of cosine, i.e., 1/cos θ.

Expressing Functions in Terms of Sine and Cosine

Many trigonometric expressions can be rewritten using only sine and cosine because these are the fundamental functions. For example, sec θ can be expressed as 1/cos θ. This helps simplify expressions by reducing them to a common set of functions.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves eliminating quotients and rewriting all terms in a consistent form, often using algebraic manipulation and identities. The goal is to express the function without fractions and only in terms of sine and cosine of θ, making it easier to analyze or solve.