This Task Appears To Be A List Of Trigonometric Expressions. Let's Organize And Format Them Properly:1. { 31 \tan 60^{\circ} $}$2. { 9 \cdot \sin 45^{\circ} \left(\tan 45^{\circ} - \cos 60^{\circ}\right) $} 3. \[ 3. \[ 3. \[ 4 \cdot \csc
Introduction
Trigonometric expressions are a fundamental part of mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will take a list of trigonometric expressions and organize and format them properly. We will then simplify each expression step by step, using various trigonometric identities and formulas.
Expression 1: 31 tan 60^{\circ}
The first expression is 31 tan 60^{\circ}. To simplify this expression, we need to recall the value of tan 60^{\circ}. Since tan 60^{\circ} is equal to √3, we can substitute this value into the expression.
31 \tan 60^{\circ} = 31 \cdot \sqrt{3}
To simplify this expression further, we can rationalize the denominator by multiplying both the numerator and the denominator by √3.
31 \cdot \sqrt{3} = \frac{31 \cdot \sqrt{3} \cdot \sqrt{3}}{\sqrt{3}}
Simplifying the numerator, we get:
\frac{31 \cdot \sqrt{3} \cdot \sqrt{3}}{\sqrt{3}} = \frac{31 \cdot 3}{\sqrt{3}}
Finally, we can simplify the expression by rationalizing the denominator.
\frac{31 \cdot 3}{\sqrt{3}} = \frac{31 \cdot 3 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}}
Simplifying the numerator and the denominator, we get:
\frac{31 \cdot 3 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{93 \cdot \sqrt{3}}{3}
Simplifying the expression further, we get:
\frac{93 \cdot \sqrt{3}}{3} = 31 \cdot \sqrt{3}
Therefore, the simplified expression is 31√3.
Expression 2: 9 \cdot \sin 45^{\circ} \left(\tan 45^{\circ} - \cos 60^{\circ}\right)
The second expression is 9 \cdot \sin 45^{\circ} \left(\tan 45^{\circ} - \cos 60^{\circ}\right). To simplify this expression, we need to recall the values of sin 45^{\circ}, tan 45^{\circ}, and cos 60^{\circ}. Since sin 45^{\circ} is equal to 1/√2, tan 45^{\circ} is equal to 1, and cos 60^{\circ} is equal to 1/2, we can substitute these values into the expression.
9 \cdot \sin 45^{\circ} \left(\tan 45^{\circ} - \cos 60^{\circ}\right) = 9 \cdot \frac{1}{\sqrt{2}} \left(1 - \frac{1}{2}\right)
Simplifying the expression inside the parentheses, we get:
9 \cdot \frac{1}{\sqrt{2}} \left(1 - \frac{1}{2}\right) = 9 \cdot \frac{1}{\sqrt{2}} \cdot \frac{1}{2}
Simplifying the expression further, we get:
9 \cdot \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{9}{2 \cdot \sqrt{2}}
To rationalize the denominator, we can multiply both the numerator and the denominator by √2.
\frac{9}{2 \cdot \sqrt{2}} = \frac{9 \cdot \sqrt{2}}{2 \cdot \sqrt{2} \cdot \sqrt{2}}
Simplifying the numerator and the denominator, we get:
\frac{9 \cdot \sqrt{2}}{2 \cdot \sqrt{2} \cdot \sqrt{2}} = \frac{9 \cdot \sqrt{2}}{2 \cdot 2}
Simplifying the expression further, we get:
\frac{9 \cdot \sqrt{2}}{2 \cdot 2} = \frac{9 \cdot \sqrt{2}}{4}
Therefore, the simplified expression is 9√2/4.
Expression 3: 4 \cdot \csc 30^{\circ} \left(\sin 30^{\circ} - \cos 45^{\circ}\right)
The third expression is 4 \cdot \csc 30^{\circ} \left(\sin 30^{\circ} - \cos 45^{\circ}\right). To simplify this expression, we need to recall the values of sin 30^{\circ}, cos 45^{\circ}, and csc 30^{\circ}. Since sin 30^{\circ} is equal to 1/2, cos 45^{\circ} is equal to 1/√2, and csc 30^{\circ} is equal to 2, we can substitute these values into the expression.
4 \cdot \csc 30^{\circ} \left(\sin 30^{\circ} - \cos 45^{\circ}\right) = 4 \cdot 2 \left(\frac{1}{2} - \frac{1}{\sqrt{2}}\right)
Simplifying the expression inside the parentheses, we get:
4 \cdot 2 \left(\frac{1}{2} - \frac{1}{\sqrt{2}}\right) = 4 \cdot 2 \cdot \frac{1}{2} \cdot \left(1 - \frac{1}{\sqrt{2}}\right)
Simplifying the expression further, we get:
4 \cdot 2 \cdot \frac{1}{2} \cdot \left(1 - \frac{1}{\sqrt{2}}\right) = 4 \cdot \left(1 - \frac{1}{\sqrt{2}}\right)
To simplify the expression further, we can multiply both the numerator and the denominator by √2.
4 \cdot \left(1 - \frac{1}{\sqrt{2}}\right) = 4 \cdot \left(\frac{\sqrt{2} - 1}{\sqrt{2}}\right)
Simplifying the expression further, we get:
4 \cdot \left(\frac{\sqrt{2} - 1}{\sqrt{2}}\right) = \frac{4 \cdot \left(\sqrt{2} - 1\right)}{\sqrt{2}}
To rationalize the denominator, we can multiply both the numerator and the denominator by √2.
\frac{4 \cdot \left(\sqrt{2} - 1\right)}{\sqrt{2}} = \frac{4 \cdot \left(\sqrt{2} - 1\right) \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}}
Simplifying the numerator and the denominator, we get:
\frac{4 \cdot \left(\sqrt{2} - 1\right) \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{4 \cdot \left(2 - \sqrt{2}\right)}{2}
Simplifying the expression further, we get:
\frac{4 \cdot \left(2 - \sqrt{2}\right)}{2} = 2 \cdot \left(2 - \sqrt{2}\right)
Therefore, the simplified expression is 2(2 - √2).
Conclusion
Q: What is the first step in simplifying a trigonometric expression?
A: The first step in simplifying a trigonometric expression is to identify the trigonometric functions involved and their respective values. This may involve recalling the values of common trigonometric functions such as sin, cos, and tan for specific angles.
Q: How do I simplify an expression with multiple trigonometric functions?
A: To simplify an expression with multiple trigonometric functions, you can use various trigonometric identities and formulas to combine the functions. For example, you can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to simplify expressions involving both sine and cosine.
Q: What is the difference between a trigonometric identity and a trigonometric formula?
A: A trigonometric identity is a statement that is always true for all values of the trigonometric function, while a trigonometric formula is a specific expression that can be used to simplify a trigonometric expression. For example, the Pythagorean identity sin^2(x) + cos^2(x) = 1 is a trigonometric identity, while the expression sin(x) = 1 - cos^2(x) is a trigonometric formula.
Q: How do I rationalize the denominator of a fraction?
A: To rationalize the denominator of a fraction, you can multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial expression a + b is a - b. For example, to rationalize the denominator of the fraction 1/√2, you can multiply both the numerator and the denominator by √2.
Q: What is the difference between a rationalized denominator and a simplified denominator?
A: A rationalized denominator is a denominator that has been multiplied by a conjugate to eliminate any radicals, while a simplified denominator is a denominator that has been simplified to its simplest form. For example, the denominator 2√2 is a rationalized denominator, while the denominator 2√2 can be simplified to 2√2 = 2√2.
Q: How do I simplify an expression with a trigonometric function and a radical?
A: To simplify an expression with a trigonometric function and a radical, you can use various trigonometric identities and formulas to combine the function and the radical. For example, you can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to simplify expressions involving both sine and cosine and radicals.
Q: What is the difference between a trigonometric expression and a trigonometric equation?
A: A trigonometric expression is a statement that involves one or more trigonometric functions, while a trigonometric equation is a statement that involves one or more trigonometric functions and an equal sign. For example, the expression sin(x) = 1 is a trigonometric expression, while the equation sin(x) = 1 is a trigonometric equation.
Q: How do I solve a trigonometric equation?
A: To solve a trigonometric equation, you can use various trigonometric identities and formulas to isolate the trigonometric function. For example, you can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to solve equations involving both sine and cosine.
Q: What is the difference between a trigonometric function and a trigonometric value?
A: A trigonometric function is a mathematical function that involves one or more trigonometric functions, while a trigonometric value is a specific value of a trigonometric function. For example, the function sin(x) is a trigonometric function, while the value sin(30°) = 1/2 is a trigonometric value.
Conclusion
In this article, we have answered some of the most frequently asked questions about simplifying trigonometric expressions. We hope that this article has been helpful in understanding how to simplify trigonometric expressions and has provided you with a better understanding of the concepts involved.