Grade 12 MathematicsQUESTION 33.1 Using Your Calculator, Evaluate The Following Function: $u(x) = 2 \sin X \cos X 3.1.1 C O M P L E T E T H E F O L L O W I N G T A B L E : 3.1.1 Complete The Following Table: 3.1.1 C O M Pl E T E T H E F O Ll O W In G T Ab L E : [ \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline X X X & − 180 ∘ -180^{\circ} − 18 0 ∘ & − 135 ∘ -135^{\circ} − 13 5 ∘ &

Introduction

In mathematics, trigonometric functions play a crucial role in various mathematical and real-world applications. The sine and cosine functions are two of the most fundamental trigonometric functions, and their product is a common occurrence in mathematical expressions. In this article, we will explore the evaluation of the function $u(x) = 2 \sin x \cos x$ using a calculator and complete a table of values for the function.

Understanding the Function

The function $u(x) = 2 \sin x \cos x$ is a product of two trigonometric functions, the sine and cosine functions. The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle, while the cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

Using a Calculator to Evaluate the Function

To evaluate the function $u(x) = 2 \sin x \cos x$ using a calculator, we need to follow these steps:

1. Enter the function: Enter the function $u(x) = 2 \sin x \cos x$ into the calculator.
2. Set the calculator mode: Set the calculator mode to degrees or radians, depending on the unit of measurement used in the problem.
3. Enter the values: Enter the values of $x$ into the calculator, and evaluate the function for each value of $x$.

Completing the Table

To complete the table, we need to evaluate the function $u(x) = 2 \sin x \cos x$ for each value of $x$ in the table.

$x$	$u(x) = 2 \sin x \cos x$
$-180^{\circ}$
$-135^{\circ}$
$-90^{\circ}$
$-45^{\circ}$
$0^{\circ}$
$45^{\circ}$
$90^{\circ}$
$135^{\circ}$
$180^{\circ}$

Calculating the Values

To calculate the values of $u(x) = 2 \sin x \cos x$ for each value of $x$ in the table, we can use the following steps:

1. Evaluate the sine and cosine functions: Evaluate the sine and cosine functions for each value of $x$ in the table.
2. Multiply the values: Multiply the values of the sine and cosine functions to obtain the value of $u(x) = 2 \sin x \cos x$.

Calculating the Value for $x = -180^{\circ}$

To calculate the value of $u(x) = 2 \sin x \cos x$ for $x = -180^{\circ}$, we need to evaluate the sine and cosine functions for $x = -180^{\circ}$.

• Evaluate the sine function: The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. For $x = -180^{\circ}$, the sine function is equal to 0.
• Evaluate the cosine function: The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. For $x = -180^{\circ}$, the cosine function is equal to -1.
• Multiply the values: Multiply the values of the sine and cosine functions to obtain the value of $u(x) = 2 \sin x \cos x$ for $x = -180^{\circ}$. $u(-180^{\circ}) = 2 \sin (-180^{\circ}) \cos (-180^{\circ}) = 2 \times 0 \times (-1) = 0$

Calculating the Value for $x = -135^{\circ}$

To calculate the value of $u(x) = 2 \sin x \cos x$ for $x = -135^{\circ}$, we need to evaluate the sine and cosine functions for $x = -135^{\circ}$.

• Evaluate the sine function: The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. For $x = -135^{\circ}$, the sine function is equal to -1.
• Evaluate the cosine function: The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. For $x = -135^{\circ}$, the cosine function is equal to -\frac{\sqrt{2}}{2}.
• Multiply the values: Multiply the values of the sine and cosine functions to obtain the value of $u(x) = 2 \sin x \cos x$ for $x = -135^{\circ}$. $u(-135^{\circ}) = 2 \sin (-135^{\circ}) \cos (-135^{\circ}) = 2 \times (-1) \times (-\frac{\sqrt{2}}{2}) = \sqrt{2}$

Calculating the Value for $x = -90^{\circ}$

To calculate the value of $u(x) = 2 \sin x \cos x$ for $x = -90^{\circ}$, we need to evaluate the sine and cosine functions for $x = -90^{\circ}$.

• Evaluate the sine function: The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. For $x = -90^{\circ}$, the sine function is equal to -1.
• Evaluate the cosine function: The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. For $x = -90^{\circ}$, the cosine function is equal to 0.
• Multiply the values: Multiply the values of the sine and cosine functions to obtain the value of $u(x) = 2 \sin x \cos x$ for $x = -90^{\circ}$. $u(-90^{\circ}) = 2 \sin (-90^{\circ}) \cos (-90^{\circ}) = 2 \times (-1) \times 0 = 0$

Calculating the Value for $x = -45^{\circ}$

To calculate the value of $u(x) = 2 \sin x \cos x$ for $x = -45^{\circ}$, we need to evaluate the sine and cosine functions for $x = -45^{\circ}$.

• Evaluate the sine function: The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. For $x = -45^{\circ}$, the sine function is equal to -\frac{\sqrt{2}}{2}.
• Evaluate the cosine function: The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. For $x = -45^{\circ}$, the cosine function is equal to -\frac{\sqrt{2}}{2}.
• Multiply the values: Multiply the values of the sine and cosine functions to obtain the value of $u(x) = 2 \sin x \cos x$ for $x = -45^{\circ}$. $u(-45^{\circ}) = 2 \sin (-45^{\circ}) \cos (-45^{\circ}) = 2 \times (-\frac{\sqrt{2}}{2}) \times (-\frac{\sqrt{2}}{2}) = 1$

Calculating the Value for $x = 0^{\circ}$

To calculate the value of $u(x) = 2 \sin x \cos x$ for $x = 0^{\circ}$, we need to evaluate the sine and cosine functions for $x = 0^{\circ}$.

• Evaluate the sine function: The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. For $x = 0^{\circ}$, the sine function is equal to 0.
• Evaluate the cosine function: The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. For $x = 0^{\circ}$, the cosine function is equal to 1.
• Multiply the values: Multiply the values of the sine and cosine functions to obtain the value of $u(x) = 2 \sin x \cos x$ for $x = 0^{\circ}$. $u(0^{\circ}) = 2 \sin (0^{\circ}) \cos (0^{\circ}) = 2 \times 0 \times 1 = 0$

Calculating the Value for $x = 45^{\circ}$

Q&A: Evaluating Trigonometric Functions

Q: What is the function $u(x) = 2 \sin x \cos x$?

A: The function $u(x) = 2 \sin x \cos x$ is a product of two trigonometric functions, the sine and cosine functions. The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle, while the cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle.

Q: How do I evaluate the function $u(x) = 2 \sin x \cos x$ using a calculator?

A: To evaluate the function $u(x) = 2 \sin x \cos x$ using a calculator, you need to follow these steps:

1. Enter the function: Enter the function $u(x) = 2 \sin x \cos x$ into the calculator.
2. Set the calculator mode: Set the calculator mode to degrees or radians, depending on the unit of measurement used in the problem.
3. Enter the values: Enter the values of $x$ into the calculator, and evaluate the function for each value of $x$.

Q: How do I complete the table of values for the function $u(x) = 2 \sin x \cos x$?

A: To complete the table of values for the function $u(x) = 2 \sin x \cos x$, you need to evaluate the function for each value of $x$ in the table.

$x$	$u(x) = 2 \sin x \cos x$
$-180^{\circ}$
$-135^{\circ}$
$-90^{\circ}$
$-45^{\circ}$
$0^{\circ}$
$45^{\circ}$
$90^{\circ}$
$135^{\circ}$
$180^{\circ}$

Q: How do I calculate the value of $u(x) = 2 \sin x \cos x$ for $x = -180^{\circ}$?

A: To calculate the value of $u(x) = 2 \sin x \cos x$ for $x = -180^{\circ}$, you need to evaluate the sine and cosine functions for $x = -180^{\circ}$.

• Evaluate the sine function: The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. For $x = -180^{\circ}$, the sine function is equal to 0.
• Evaluate the cosine function: The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. For $x = -180^{\circ}$, the cosine function is equal to -1.
• Multiply the values: Multiply the values of the sine and cosine functions to obtain the value of $u(x) = 2 \sin x \cos x$ for $x = -180^{\circ}$. $u(-180^{\circ}) = 2 \sin (-180^{\circ}) \cos (-180^{\circ}) = 2 \times 0 \times (-1) = 0$

Q: How do I calculate the value of $u(x) = 2 \sin x \cos x$ for $x = -135^{\circ}$?

A: To calculate the value of $u(x) = 2 \sin x \cos x$ for $x = -135^{\circ}$, you need to evaluate the sine and cosine functions for $x = -135^{\circ}$.

• Evaluate the sine function: The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. For $x = -135^{\circ}$, the sine function is equal to -1.
• Evaluate the cosine function: The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. For $x = -135^{\circ}$, the cosine function is equal to -\frac{\sqrt{2}}{2}.
• Multiply the values: Multiply the values of the sine and cosine functions to obtain the value of $u(x) = 2 \sin x \cos x$ for $x = -135^{\circ}$. $u(-135^{\circ}) = 2 \sin (-135^{\circ}) \cos (-135^{\circ}) = 2 \times (-1) \times (-\frac{\sqrt{2}}{2}) = \sqrt{2}$

Q: How do I calculate the value of $u(x) = 2 \sin x \cos x$ for $x = -90^{\circ}$?

A: To calculate the value of $u(x) = 2 \sin x \cos x$ for $x = -90^{\circ}$, you need to evaluate the sine and cosine functions for $x = -90^{\circ}$.

• Evaluate the sine function: The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. For $x = -90^{\circ}$, the sine function is equal to -1.
• Evaluate the cosine function: The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. For $x = -90^{\circ}$, the cosine function is equal to 0.
• Multiply the values: Multiply the values of the sine and cosine functions to obtain the value of $u(x) = 2 \sin x \cos x$ for $x = -90^{\circ}$. $u(-90^{\circ}) = 2 \sin (-90^{\circ}) \cos (-90^{\circ}) = 2 \times (-1) \times 0 = 0$

Q: How do I calculate the value of $u(x) = 2 \sin x \cos x$ for $x = -45^{\circ}$?

A: To calculate the value of $u(x) = 2 \sin x \cos x$ for $x = -45^{\circ}$, you need to evaluate the sine and cosine functions for $x = -45^{\circ}$.

• Evaluate the sine function: The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. For $x = -45^{\circ}$, the sine function is equal to -\frac{\sqrt{2}}{2}.
• Evaluate the cosine function: The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. For $x = -45^{\circ}$, the cosine function is equal to -\frac{\sqrt{2}}{2}.
• Multiply the values: Multiply the values of the sine and cosine functions to obtain the value of $u(x) = 2 \sin x \cos x$ for $x = -45^{\circ}$. $u(-45^{\circ}) = 2 \sin (-45^{\circ}) \cos (-45^{\circ}) = 2 \times (-\frac{\sqrt{2}}{2}) \times (-\frac{\sqrt{2}}{2}) = 1$

Q: How do I calculate the value of $u(x) = 2 \sin x \cos x$ for $x = 0^{\circ}$?

A: To calculate the value of $u(x) = 2 \sin x \cos x$ for $x = 0^{\circ}$, you need to evaluate the sine and cosine functions for $x = 0^{\circ}$.

• Evaluate the sine function: The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. For $x = 0^{\circ}$, the sine function is equal to 0.
• Evaluate the cosine function: The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. For $x = 0^{\circ}$, the cosine function is equal to 1.
• Multiply the values: Multiply the values of the sine and cosine functions to obtain the value of $u(x) = 2 \sin x \cos x$ for $x = 0^{\circ}$. $u(0^{\circ}) = 2 \sin (0^{\circ}) \cos (0^{\circ}) = 2 \times 0 \times 1 = 0$

**Q: How do I calculate the value of $u(x) = 2 \sin x \cos x$ for $