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mathematics UK Year 13 Mathematics (A Levels)

Solved problems for the chapter:

Graphing Trigonometric Functions Transformations

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Graph of tangent and cotangent functions

1. Graphs of the form f (x) = A tan (x), g (x) = A cot (x)

2. Graphs of the form f (x) = tan (k x), g (x) = cot (k x)

2.1. Period of tangent and cotangent functions

3. Graphs of the form f (x) = tan (x) + n, f (x) = cot (x) + n

4. Graphs of the form f (x) = tan (x - a), g (x) = cot (x - a)

5. Graphs of the form f (x) = A tan (x - a) + n, g (x) = A cot (x - a) + n

6. Forum

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Applications of the derivative

1. Determination of tangent and normal to a curve:

1.1. Determining the tangent to a curve:

1.2. Determining the normal to a curve:

2. Using differentiation to calculate angles:

2.1. Angle between a curve and the abscissa axis:

2.2. Angle between a curve and the ordinate axis:

2.3. Angle between two curves

3. Determining the rise and fall of a function and its stationary points

3.1. Increasing and decreasing function:

3.2. Stationary points:

4. Extreme problems:

5. Forum

↓ Show theory [2] ↓

Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:graphing_trigonometric_functions_transformations/4940/

Example 1.

Draw a graph of the function , .

Calculate the zeros of the function.
At what intervals is the function negative?

Example 2.

The function is given.

Calculate accurately
Write down the zeros and intercepts of the function and draw its graph

Draw a graph of the function:

Draw a graph of the functions and . Also record the period and range of the values of the functions and .

Choose the correct answer in each task:

Find the value of .
Simplify .
The derivative of the function .
Simplify , .

The table of values of a quadratic function is tabulated as shown below:

x	- 2	- 1	0	1	2	3	4
f(x)	- 5	0	3	4	3	0	-5

Read and write down from the table:
- the zeros
- the vertex
- the y-intersection
- graph the function in the given coordinate system.
Obtain the equation of the quadratic function f.
Obtain the equation of the tangent to the graph of the quadratic function

at the point .

Given a circle with radius 6 m and centre O. The points and are drawn on the circle so that the angles and are right angles (as shown in the diagram below for one of the possible drawn situations). The size of the angle depends on the selection of the positions of the points and on the circle.

Prove that for any points and drawn in this way, the area of the shaded area as a function of the size of the angle is equal to , if the angle is expressed in radians.
For which obtuse angle is the area of the shaded area the largest? Round the angle to the nearest tenth of a degree.
The diagram above also shows the cross-section of a tunnel with a plane perpendicular to the direction of travel. The circle has a radius of , the width of the tunnel is , and its length is . The curved ceiling and walls of the tunnel will be painted over. For this purpose, 400 l of paint was bought. Will there be enough paint if they use exactly 5 l of paint for ? Justify your answer.

10.

Draw the points A, B, C and D in the coordinate system below and obtain the equation of the triangle of the outlined circle.
Calculate the area of quadrilateral ABCD
Rotate the triangle ABC by around the side AC to get the geometric shape G. Prove that the volume of the solid G is , where and .
Calculate the abscissa of the point T on the x-axis so that the sum of the squares of the distances from the point T to the point A and from the point T to the point C is minimal.

11.

Given four trapeziums ABCD where the bases and , and the length of the side .

In the first trapezium, the non-parallel sides intersect at the point E. Calculate the length of the distance DE.
In the second trapezium, the size of the angle is equal to . Calculate the size of the angle . Round the result to the nearest minute.
In the third trapezium, the diagonal BD measures . Calculate the length of .
In the fourth trapezium, the size of the angle is equal to . There is a point T on the side AD and a point V on the side AB such that . Calculate at what length the area of the triangle VBT will be maximum.

12.

The function f is given by .

Plot the graph of the function f.
Obtain the equation of the tangent to the function f at the point with abscissa .

13.

A quadrilateral is orthodiagonal if its diagonals intersect at right angles. The area of any orthodiagonal quadrilateral with diagonal e and f is given as .

Let be such points on the unit circle that the quadrilateral ABCD is a trapezium with bases AB and CD. The diagonals of the trapezium intersect at the point P.

Let O be the origin of the coordinate system and be the size of the angle AOB.

Prove that trapezium ABCD is isosceles. Draw a sketch.
Calculate the sizes of the angles and and prove that the trapezium ABCD is orthodiagonal.
Express the coordinates of the point B and the length of the diagonal in term of .
Express the area of the trapezium ABCD in term of . Which point B on the unit circle will give the maximum area of the trapezium ABCD? Justify your answer.