Evaluate The Expression:${ \frac{\sin 30^{\circ} + \cos 60^{\circ} + \csc 30^{\circ}}{\sin 45^{\circ} \cdot \sin 30^{\circ} + \tan 45^{\circ}} }$

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will evaluate the expression $\frac{\sin 30^{\circ} + \cos 60^{\circ} + \csc 30^{\circ}}{\sin 45^{\circ} \cdot \sin 30^{\circ} + \tan 45^{\circ}}$ using various trigonometric identities and formulas.

Understanding the Components of the Expression

Before we dive into the evaluation of the expression, let's break down its components and understand their values.

• $\sin 30^{\circ}$: This is the sine of 30 degrees, which is equal to $\frac{1}{2}$.
• $\cos 60^{\circ}$: This is the cosine of 60 degrees, which is equal to $\frac{1}{2}$.
• $\csc 30^{\circ}$: This is the cosecant of 30 degrees, which is equal to $\frac{1}{\sin 30^{\circ}} = 2$.
• $\sin 45^{\circ}$: This is the sine of 45 degrees, which is equal to $\frac{\sqrt{2}}{2}$.
• $\tan 45^{\circ}$: This is the tangent of 45 degrees, which is equal to $1$.

Evaluating the Expression

Now that we have understood the components of the expression, let's evaluate it step by step.

Step 1: Simplify the Numerator

The numerator of the expression is $\sin 30^{\circ} + \cos 60^{\circ} + \csc 30^{\circ}$. We can simplify this by substituting the values of the trigonometric functions.

$\sin 30^{\circ} + \cos 60^{\circ} + \csc 30^{\circ} = \frac{1}{2} + \frac{1}{2} + 2 = 3$

Step 2: Simplify the Denominator

The denominator of the expression is $\sin 45^{\circ} \cdot \sin 30^{\circ} + \tan 45^{\circ}$. We can simplify this by substituting the values of the trigonometric functions.

$\sin 45^{\circ} \cdot \sin 30^{\circ} + \tan 45^{\circ} = \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + 1 = \frac{\sqrt{2}}{4} + 1$

Step 3: Simplify the Expression

Now that we have simplified the numerator and denominator, we can simplify the expression by dividing the numerator by the denominator.

$\frac{\sin 30^{\circ} + \cos 60^{\circ} + \csc 30^{\circ}}{\sin 45^{\circ} \cdot \sin 30^{\circ} + \tan 45^{\circ}} = \frac{3}{\frac{\sqrt{2}}{4} + 1}$

Rationalizing the Denominator

To rationalize the denominator, we can multiply the numerator and denominator by the conjugate of the denominator.

$\frac{3}{\frac{\sqrt{2}}{4} + 1} = \frac{3}{\frac{\sqrt{2}}{4} + \frac{4}{4}} = \frac{3}{\frac{4 + \sqrt{2}}{4}} = \frac{3 \cdot 4}{4 + \sqrt{2}} = \frac{12}{4 + \sqrt{2}}$

Simplifying the Expression

Now that we have rationalized the denominator, we can simplify the expression by multiplying the numerator and denominator by the conjugate of the denominator.

$\frac{12}{4 + \sqrt{2}} = \frac{12}{4 + \sqrt{2}} \cdot \frac{4 - \sqrt{2}}{4 - \sqrt{2}} = \frac{12(4 - \sqrt{2})}{16 - 2} = \frac{48 - 12\sqrt{2}}{14}$

Conclusion

In this article, we evaluated the expression $\frac{\sin 30^{\circ} + \cos 60^{\circ} + \csc 30^{\circ}}{\sin 45^{\circ} \cdot \sin 30^{\circ} + \tan 45^{\circ}}$ using various trigonometric identities and formulas. We simplified the numerator and denominator, rationalized the denominator, and finally simplified the expression to $\frac{48 - 12\sqrt{2}}{14}$.

Final Answer

The final answer is $\boxed{\frac{48 - 12\sqrt{2}}{14}}$.

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Introduction

In our previous article, we evaluated the expression $\frac{\sin 30^{\circ} + \cos 60^{\circ} + \csc 30^{\circ}}{\sin 45^{\circ} \cdot \sin 30^{\circ} + \tan 45^{\circ}}$ using various trigonometric identities and formulas. In this article, we will answer some frequently asked questions related to the evaluation of this expression.

Q&A

Q1: What is the value of $\sin 30^{\circ}$?

A1: The value of $\sin 30^{\circ}$ is $\frac{1}{2}$.

Q2: What is the value of $\cos 60^{\circ}$?

A2: The value of $\cos 60^{\circ}$ is $\frac{1}{2}$.

Q3: What is the value of $\csc 30^{\circ}$?

A3: The value of $\csc 30^{\circ}$ is $2$.

Q4: What is the value of $\sin 45^{\circ}$?

A4: The value of $\sin 45^{\circ}$ is $\frac{\sqrt{2}}{2}$.

Q5: What is the value of $\tan 45^{\circ}$?

A5: The value of $\tan 45^{\circ}$ is $1$.

Q6: How do we simplify the numerator of the expression?

A6: We simplify the numerator by substituting the values of the trigonometric functions. In this case, the numerator is $\sin 30^{\circ} + \cos 60^{\circ} + \csc 30^{\circ}$. We can simplify this by substituting the values of the trigonometric functions.

Q7: How do we simplify the denominator of the expression?

A7: We simplify the denominator by substituting the values of the trigonometric functions. In this case, the denominator is $\sin 45^{\circ} \cdot \sin 30^{\circ} + \tan 45^{\circ}$. We can simplify this by substituting the values of the trigonometric functions.

Q8: How do we rationalize the denominator?

A8: We rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

Q9: What is the final answer to the expression?

A9: The final answer to the expression is $\boxed{\frac{48 - 12\sqrt{2}}{14}}$.

Conclusion

In this article, we answered some frequently asked questions related to the evaluation of the expression $\frac{\sin 30^{\circ} + \cos 60^{\circ} + \csc 30^{\circ}}{\sin 45^{\circ} \cdot \sin 30^{\circ} + \tan 45^{\circ}}$. We provided the values of the trigonometric functions, simplified the numerator and denominator, rationalized the denominator, and finally simplified the expression to $\boxed{\frac{48 - 12\sqrt{2}}{14}}$.

Final Answer

The final answer is $\boxed{\frac{48 - 12\sqrt{2}}{14}}$.