Problem 17 - Algebra 2 and Trigonometry

Question

In \(15-22,\) write each given expression in terms of sine and cosine and express the result in simplest form. \((\tan \theta)(\cos \theta)\)

Step-by-Step Solution

Answer

The simplified expression is \(\sin \theta\).

Step 1: Recall the Trigonometric Identity for Tangent

Recall that the tangent function can be expressed in terms of sine and cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

Step 2: Substitute and Simplify the Expression

Substitute the expression for \(\tan \theta\) into the original expression \((\tan \theta)(\cos \theta)\) so that it becomes \(\left( \frac{\sin \theta}{\cos \theta} \right)(\cos \theta)\).

Step 3: Simplify Further by Canceling Terms

Cancel the \(\cos \theta\) term from the numerator and the denominator, resulting in the expression \(\sin \theta\).

Step 4: Express the Final Result

The expression \((\tan \theta)(\cos \theta)\) simplifies to \(\sin \theta\) using the relationships between trigonometric functions.

Key Concepts

sine and cosine

Trigonometric functions such as sine and cosine are foundational to trigonometry. They define the relationship between angles and side lengths in right-angled triangles.

The cosine of an angle, \(\cos \theta\), is the ratio of the length of the adjacent side to the hypotenuse. This gives us a measure of how distant the angle's opposite side is from being perpendicular to the hypotenuse.

Similarly, the sine of an angle, \(\sin \theta\), represents the ratio of the length of the opposite side to the hypotenuse.

• Sine: \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\)
• Cosine: \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\)

These relationships are particularly useful in converting trigonometric expressions into useful forms. **Understanding sine and cosine is crucial as they are the building blocks of more complex trigonometric identities,** and enable us to express one function in terms of another.

tangent function

The tangent function connects directly with sine and cosine, providing another layer to the relationships between trigonometric ratios. It is defined as the ratio of sine to cosine:

\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]
This identity indicates that the tangent function can be broken down into simpler components, using the basic sine and cosine functions. Because of this, **the tangent is often used in simplifying expressions involving these functions.**

When expressing a trigonometric expression, recognizing \(\tan \theta\) as a function of \(\sin \theta\) and \(\cos \theta\) allows for the reduction and simplification of complex trigonometric terms. This makes trigonometry not only more intuitive but also more computationally efficient. **The tangent's reliance on sine and cosine accounts for its utility in trigonometric identities and simplifies transformations like the one in the exercise.**

simplifying expressions

Simplifying expressions is a crucial skill when working with trigonometric functions. By writing functions in terms of simpler functions like sine and cosine, equations become easier to understand and solve. This is notably useful when dealing with complex or compound expressions.

Consider an expression initially involving multiple trigonometric functions. Breaking them down into basic components allows us to cancel terms and simplify. For example, in the expression \(\left( \frac{\sin \theta}{\cos \theta} \right)(\cos \theta)\), recognizing common factors (like \(\cos \theta\)) allows us to eliminate them and obtain a simpler form, \(\sin \theta\).

• Identifying and replacing complex terms with basic trigonometric identities, such as tangent with sine over cosine, is a key step.
• After substitution, use mathematical operations like factoring and canceling to reduce the expression further.
• Always recheck your final expression to ensure that it is in the simplest form possible.

Through practice, simplifying expressions becomes an intuitive process, enhancing both understanding and computation speed in trigonometry.