Simplify The Expression: 1 Sin ⁡ 30 ∘ + Cos ⁡ 2 30 ∘ 1 \sin 30^{\circ} + \cos^2 30^{\circ} 1 Sin 3 0 ∘ + Cos 2 3 0 ∘

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Introduction

In this article, we will simplify the given trigonometric expression: $1 \sin 30^{\circ} + \cos^2 30^{\circ}$. This expression involves the sine and cosine functions, which are fundamental concepts in trigonometry. We will use various trigonometric identities and formulas to simplify the expression and arrive at the final result.

Understanding the Expression

The given expression is $1 \sin 30^{\circ} + \cos^2 30^{\circ}$. This expression involves two terms: $1 \sin 30^{\circ}$ and $\cos^2 30^{\circ}$. The first term is the product of 1 and the sine of 30 degrees, while the second term is the square of the cosine of 30 degrees.

Recalling Trigonometric Identities

To simplify the expression, we need to recall some basic trigonometric identities. One of the most important identities is the Pythagorean identity, which states that $\sin^2 \theta + \cos^2 \theta = 1$ for any angle $\theta$. We will use this identity to simplify the expression.

Simplifying the Expression

Let's start by simplifying the first term: $1 \sin 30^{\circ}$. Since the sine of 30 degrees is equal to 0.5, we can rewrite the first term as:

$1 \sin 30^{\circ} = 1 \times 0.5 = 0.5$

Now, let's simplify the second term: $\cos^2 30^{\circ}$. Using the Pythagorean identity, we know that $\sin^2 \theta + \cos^2 \theta = 1$. Since the sine of 30 degrees is 0.5, we can rewrite the Pythagorean identity as:

$\sin^2 30^{\circ} + \cos^2 30^{\circ} = 1$ $(0.5)^2 + \cos^2 30^{\circ} = 1$ $0.25 + \cos^2 30^{\circ} = 1$

Now, let's isolate the $\cos^2 30^{\circ}$ term:

$\cos^2 30^{\circ} = 1 - 0.25$ $\cos^2 30^{\circ} = 0.75$

Now that we have simplified both terms, we can rewrite the original expression as:

$1 \sin 30^{\circ} + \cos^2 30^{\circ} = 0.5 + 0.75$

Final Result

Now, let's add the two terms together:

$0.5 + 0.75 = 1.25$

Therefore, the simplified expression is $1.25$.

Conclusion

In this article, we simplified the given trigonometric expression: $1 \sin 30^{\circ} + \cos^2 30^{\circ}$. We used various trigonometric identities and formulas to arrive at the final result. The simplified expression is $1.25$. We hope this article has provided a clear understanding of how to simplify trigonometric expressions.

Additional Tips and Tricks

• When simplifying trigonometric expressions, always recall the basic trigonometric identities, such as the Pythagorean identity.
• Use algebraic manipulations to isolate the trigonometric terms.
• Simplify the trigonometric terms using the trigonometric identities.
• Check the final result by plugging it back into the original expression.

Common Mistakes to Avoid

• Not recalling the basic trigonometric identities.
• Not using algebraic manipulations to isolate the trigonometric terms.
• Not simplifying the trigonometric terms using the trigonometric identities.
• Not checking the final result by plugging it back into the original expression.

Real-World Applications

Trigonometric expressions are used in various real-world applications, such as:

• Navigation: Trigonometric expressions are used to calculate distances and angles in navigation.
• Physics: Trigonometric expressions are used to describe the motion of objects in physics.
• Engineering: Trigonometric expressions are used to design and analyze complex systems in engineering.

Final Thoughts

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Introduction

In our previous article, we simplified the given trigonometric expression: $1 \sin 30^{\circ} + \cos^2 30^{\circ}$. We used various trigonometric identities and formulas to arrive at the final result. In this article, we will answer some frequently asked questions related to the simplification of trigonometric expressions.

Q&A

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that states $\sin^2 \theta + \cos^2 \theta = 1$ for any angle $\theta$.

Q: How do I simplify a trigonometric expression?

A: To simplify a trigonometric expression, recall the basic trigonometric identities, use algebraic manipulations to isolate the trigonometric terms, and simplify the trigonometric terms using the trigonometric identities.

Q: What is the difference between $\sin \theta$ and $\cos \theta$?

A: $\sin \theta$ represents the ratio of the opposite side to the hypotenuse in a right-angled triangle, while $\cos \theta$ represents the ratio of the adjacent side to the hypotenuse.

Q: How do I use the Pythagorean identity to simplify a trigonometric expression?

A: To use the Pythagorean identity to simplify a trigonometric expression, recall the identity and substitute the values of the trigonometric functions into the identity.

Q: What is the value of $\sin 30^{\circ}$?

A: The value of $\sin 30^{\circ}$ is 0.5.

Q: What is the value of $\cos 30^{\circ}$?

A: The value of $\cos 30^{\circ}$ is $\frac{\sqrt{3}}{2}$.

Q: How do I simplify the expression $1 \sin 30^{\circ} + \cos^2 30^{\circ}$?

A: To simplify the expression $1 \sin 30^{\circ} + \cos^2 30^{\circ}$, recall the Pythagorean identity and substitute the values of the trigonometric functions into the identity.

Q: What is the final result of the expression $1 \sin 30^{\circ} + \cos^2 30^{\circ}$?

A: The final result of the expression $1 \sin 30^{\circ} + \cos^2 30^{\circ}$ is 1.25.

Common Mistakes to Avoid

• Not recalling the basic trigonometric identities.
• Not using algebraic manipulations to isolate the trigonometric terms.
• Not simplifying the trigonometric terms using the trigonometric identities.
• Not checking the final result by plugging it back into the original expression.

Real-World Applications

Trigonometric expressions are used in various real-world applications, such as:

• Navigation: Trigonometric expressions are used to calculate distances and angles in navigation.
• Physics: Trigonometric expressions are used to describe the motion of objects in physics.
• Engineering: Trigonometric expressions are used to design and analyze complex systems in engineering.

Final Thoughts

In conclusion, simplifying trigonometric expressions is an essential skill in mathematics. By recalling the basic trigonometric identities and using algebraic manipulations, we can simplify complex trigonometric expressions. We hope this article has provided a clear understanding of how to simplify trigonometric expressions and has inspired readers to explore the world of trigonometry.

Additional Resources

• Khan Academy: Trigonometry
• MIT OpenCourseWare: Trigonometry
• Wolfram Alpha: Trigonometry

Conclusion

In this article, we answered some frequently asked questions related to the simplification of trigonometric expressions. We hope this article has provided a clear understanding of how to simplify trigonometric expressions and has inspired readers to explore the world of trigonometry.