Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-26,\) prove that each equation is an identity. $$ \tan \theta \sin \theta \cos \theta=\sin ^{2} \theta $$

Step-by-Step Solution

Answer

The equation is an identity because both sides equal \( \sin^2 \theta \).

Step 1: Understanding trigonometric functions

To prove the given equation is an identity, first understand that \( \tan \theta \) can be expressed as \( \frac{\sin \theta}{\cos \theta} \).

Step 2: Substitute and Simplify

Substitute \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) into the left side of the equation: \( \frac{\sin \theta}{\cos \theta} \cdot \sin \theta \cdot \cos \theta \). Simplifying this, you get \( \sin \theta \cdot 1 \), which is \( \sin ^{2} \theta \).

Step 3: Compare LHS and RHS

Now, we have simplified the left side (LHS) to be \( \sin ^{2} \theta \) and the right side (RHS) is already given as \( \sin ^{2} \theta \). Both sides are equal.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are fundamental in trigonometry and widely used in various fields like physics, engineering, and architecture. In our context, some of the primary functions you will encounter include sine (\(\sin \theta\)), cosine (\(\cos \theta\)), and tangent (\(\tan \theta\)).

Each of these functions has a specific relationship in terms of a right-angled triangle.

• \(\sin \theta\) is the ratio of the opposite side to the hypotenuse.
• \(\cos \theta\) is the ratio of the adjacent side to the hypotenuse.
• \(\tan \theta\) is the ratio of the opposite side to the adjacent side or mathematically, \(\frac{\sin \theta}{\cos \theta}\).

Understanding these basic definitions allows you to unravel more complex trigonometric identities. When proving identities, it is crucial to remember these fundamental relationships and manipulatively express them to simplify and solve equations.

Simplifying Expressions

Simplifying expressions is an essential skill in solving mathematical problems, especially in proving trigonometric identities. Simplifying involves breaking down complicated expressions into easier, more manageable parts, and using algebraic rules to reduce them.

In the given exercise, the expression \(\tan \theta \sin \theta \cos \theta\) needs simplification.

• First, identify and substitute equivalent expressions, such as converting \(\tan \theta\) to \(\frac{\sin \theta}{\cos \theta}\).
• After substitution, the expression becomes \(\frac{\sin \theta}{\cos \theta} \cdot \sin \theta \cdot \cos \theta\).
• Proceed by simplifying step-by-step which involves canceling out terms or reducing fractions, in this case, \(\cos \theta\) cancels out, leaving \(\sin \theta \cdot 1\).

By understanding which terms can be canceled and rewriting expressions, the expression is simplified to \(\sin^2 \theta\), aligning both sides of the equation effectively.

Proof Strategies

When proving identities, having a structured strategy is key. Begin by understanding the identity you want to prove and then work systematically to show each step.

Proof in trigonometry often incorporates logical reasoning and algebraic manipulation.

• Start by expressing different trigonometric functions in their basic forms or in terms of one another, such as \(\tan \theta\) as \(\frac{\sin \theta}{\cos \theta}\).
• Next, simplify the left-hand side (LHS) or right-hand side (RHS) of the expression. This may involve canceling out similar terms or utilizing other known identities.
• Compare both sides of the equation; the goal is to show they are equivalent, thus proving the identity.

For example, once the LHS of the equation simplifies to \(\sin^2 \theta\), one can immediately note that it matches the RHS. By employing a mix of algebraic operations and recalling basic identities, you can efficiently reach your conclusion and affirm that the given equation holds true as an identity.