NSC Technical Mathematics Trigonometry: Given the functions defined by

Given the functions defined by $f(x) = ext{sin } x$ and g(x) = ext{cos } 2x, ext{ where } x ext{ in } [0^{ heta}; 180^{ heta}] 5.1 Write down the period of g - NSC Technical Mathematics - Question 5 - 2022 - Paper 2

Question 5

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Given the functions defined by $f(x) = ext{sin } x$ and g(x) = ext{cos } 2x, ext{ where } x ext{ in } [0^{ heta}; 180^{ heta}] 5.1 Write down the period ... show full transcript

Worked Solution & Example Answer:Given the functions defined by $f(x) = ext{sin } x$ and g(x) = ext{cos } 2x, ext{ where } x ext{ in } [0^{ heta}; 180^{ heta}] 5.1 Write down the period of g - NSC Technical Mathematics - Question 5 - 2022 - Paper 2

Step 1

5.1 Write down the period of g.

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Answer

The period of the function $g(x) = \cos{2x}$ can be found using the formula for the period of cosine functions, which is given by:

$ext{Period} = \frac{2\pi}{|b|}$

In this case, $b = 2$ , so:

$ext{Period} = \frac{2\pi}{2} = \pi$

However, since we are asked for the period in degrees:

$\pi ext{ radians} = 180^{\circ}$

Thus, the period of $g$ is $180^{\circ}$ .

Step 2

5.2 Draw sketch graphs of f and g on the same set of axes on the grid provided in the ANSWER BOOK.

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Answer

To sketch the graphs of $f(x) = \sin{x}$ and $g(x) = \cos{2x}$ :

Graph of $f(x) = \sin{x}$ :
- The sine function oscillates between -1 and 1.
- It has a period of $360^{\circ}$ .
- The turning points occur at $x = 0^{\circ}, 90^{\circ}, 180^{\circ}, 270^{\circ}, 360^{\circ}$ with corresponding $y$ values, respectively, being $0, 1, 0, -1, 0$ .
- The graph will start at the origin, peak at $90^{\circ}$ , cross at $180^{\circ}$ , trough at $270^{\circ}$ , and return to the axis at $360^{\circ}$ .
Graph of $g(x) = \cos{2x}$ :
- This cosine function also oscillates between -1 and 1 but has a period of $180^{\circ}$ , due to the factor of 2.
- The key points occur at $0^{\circ}, 90^{\circ}, 180^{\circ}$ , with the values $1, 0, -1$ at these points, respectively.
- The graph starts at its maximum, crosses the axis at $90^{\circ}$ , and reaches its minimum at $180^{\circ}$ .

Make sure to label all turning points, endpoints, and intercepts on the axes.

Given the functions defined by $f(x) = ext{sin } x$ and g(x) = ext{cos } 2x, ext{ where } x ext{ in } [0^{ heta}; 180^{ heta}] 5.1 Write down the period of g - NSC Technical Mathematics - Question 5 - 2022 - Paper 2

Worked Solution & Example Answer:Given the functions defined by $f(x) = ext{sin } x$ and g(x) = ext{cos } 2x, ext{ where } x ext{ in } [0^{ heta}; 180^{ heta}] 5.1 Write down the period of g - NSC Technical Mathematics - Question 5 - 2022 - Paper 2

5.1 Write down the period of g.

5.2 Draw sketch graphs of f and g on the same set of axes on the grid provided in the ANSWER BOOK.

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