Question

Simplify \(\sin \theta \cos \theta \tan \theta+\cos ^{2} \theta\)

Short answer

Answer: The simplified form of the given expression is \(1\).

Step-by-step solution

Express everything in terms of sin and cos

Recall that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), so we can replace the \(\tan \theta\) in our expression. The expression then becomes: \[ \sin \theta \cos \theta \frac{\sin \theta}{\cos \theta}+\cos ^{2} \theta \]

Simplify the expression

Now we can simplify the expression by canceling out terms and rearranging. Indeed: \[ \sin \theta \cos \theta \frac{\sin \theta}{\cos \theta}+\cos ^{2} \theta = \sin \theta \cos \theta \cdot \frac{\sin \theta}{\cancel{\cos \theta}}\cancel{\cos \theta}+\cos ^{2} \theta := \sin^{2} \theta +\cos ^{2} \theta\]

Recognize the Pythagorean identity

Recall the Pythagorean identity for sin and cos: \[ \sin^{2} \theta + \cos^{2} \theta = 1\]

Final simplification

Since our simplified expression is equal to the Pythagorean identity, the final simplified result is: \[1\]

Key concepts

Pythagorean Identity

The Pythagorean Identity is one of the most fundamental and widely used identities in trigonometry. It states that for any angle \(\theta\), the identity is given as:\[ \sin^2 \theta + \cos^2 \theta = 1 \]This identity comes from the Pythagorean Theorem applied to the unit circle, where the hypotenuse is always 1. It demonstrates the relationship between the sine and cosine of an angle. This is particularly useful because it helps to simplify trigonometric expressions by allowing part of the expression to be rewritten or combined differently. In our exercise, when we simplified the expression \(\sin^2 \theta + \cos^2 \theta\), we directly applied the Pythagorean Identity. The end result, thanks to this identity, was simply 1. This shows the power of this identity in solving and simplifying complex trigonometric problems.

• Useful in simplifying expressions
• Directly relates sine and cosine functions
• Derived from the geometry of a circle

It's important to become familiar with this identity as it will come in handy in various trigonometric tasks.

Simplification of Trigonometric Expressions

Simplification is a crucial skill in solving trigonometric problems, where often complex expressions can be reduced to simpler forms for easy evaluation. In our exercise, we simplified the expression by substituting and cancelling terms. We started with \(\sin \theta \cos \theta \tan \theta + \cos^2 \theta\). By replacing \(\tan \theta\) with \(\frac{\sin \theta}{\cos \theta}\), the expression became easier to manage.Next, we noticed that the \(\cos \theta\) terms in both the numerator and denominator of the first part of the expression were cancellable, which simplified the expression further. This left us with \(\sin^2 \theta + \cos^2 \theta\). This process of substituting, cancelling, and reordering terms is common in simplifying trigonometric expressions.

• Replace variables using known identities
• Cancel terms where possible to simplify
• Recognize familiar patterns, like the Pythagorean Identity, to further simplify

By practicing these techniques, you'll find that simplifying trigonometric expressions becomes more intuitive and systematic.

Trigonometric Functions

Trigonometric functions such as sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)) are foundational in the study of geometry and periodic phenomena. Let's break them down:- The sine of an angle \(\theta\) measures the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. It is denoted by \(\sin \theta\).- The cosine of an angle \(\theta\) is the ratio of the length of the adjacent side to the hypotenuse, expressed as \(\cos \theta\).- The tangent of an angle is the ratio of the sine of the angle to its cosine, defined as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).These functions aren't just vital in theoretical math—they’re used to model real-world situations like sound waves and tides. In the context of simplification, knowing these definitions allows us to swap or cancel functions to simplify expressions, just like we did in the exercise. Understanding these basic functions is key to mastering trigonometry.

• Sine and cosine describe relationships in triangles and waves
• Tangent relates to the slope associated with an angle
• Critical in expression substitution and simplification

Familiarity with these functions lays the groundwork for exploring more complex trigonometric concepts and relationships.